Healing Intelligence: A Bio-Inspired Metaheuristic Optimization Method Using Recovery Dynamics

Abstract

BioHealing Optimization (BHO) is a bio-inspired metaheuristic that operationalizes the injury–recovery paradigm through an iterative loop of recombination, stochastic injury, and guided healing. The algorithm is further enhanced by adaptive mechanisms, including scar map, hot-dimension focusing, RAGE/hyper-RAGE bursts (Rapid Aggressive Global Exploration), and healing-rate modulation, enabling a dynamic balance between exploration and exploitation. Across 17 benchmark problems with 30 runs, each under a fixed budget of $1.5·{10}^5$ function evaluations, BHO achieves the lowest overall rank in both the “best-of-runs” (47) and the “mean-of-runs” (48), giving an overall rank sum of 95 and an average rank of 2.794. Representative first-place results include Frequency-Modulated Sound Waves, the Lennard–Jones potential, and Electricity Transmission Pricing. In contrast to prior healing-inspired optimizers such as Wound Healing Optimization (WHO) and Synergistic Fibroblast Optimization (SFO), BHO uniquely integrates (i) an explicit tri-phasic architecture (DE/best/1/bin recombination → Gaussian/Lévy injury → guided healing), (ii) per-dimension stateful adaptation (scar map, hot-dims), and (iii) stagnation-triggered bursts (RAGE/hyper-RAGE). These features provide a principled exploration–exploitation separation that is absent in WHO/SFO.

1. Introduction

Global optimization refers to the task of identifying the global minimum of a real-valued, continuous objective function $f\left(x\right)$, where the variable x belongs to a predefined, bounded search space $S\subset {\mathbb{R}}^n$. The goal is to determine the point $x^{*}\in S$ such that the function $f\left(x\right)$ achieves its lowest possible value over the entire domain:

$$x^{*}=arg\underset{x\in S}{min}f\left(x\right)$$

(1)

where $f\left(x\right)$ is the objective function to be minimized. This function can represent a variety of criteria depending on the problem context, such as cost, loss, error, potential energy, or any other performance metric. S is the feasible search space, a compact subset of ${\mathbb{R}}^n$, meaning it is both closed and bounded. Typically, S is defined as an n-dimensional hyperrectangle (also called a box constraint), given by

$$S=\left[a_1,b_1\right]\otimes \left[a_2,b_2\right]\otimes \dots \left[a_n,b_n\right]$$

This denotes that each variable $x_i$ is constrained within a finite interval: $x_i\in [a_i,b_i],\mathrm{for}i=1,2,\dots ,n$. The Cartesian product of these intervals defines the multidimensional region where the search for the global optimum takes place.

Optimization is one of the most fundamental and widely applied domains of computational intelligence, with a vast range of applications in scientific, technological, and industrial fields. Although classical optimization techniques can be effective for small- or medium-scale problems, they often fail to deliver satisfactory results when applied to complex, nonlinear, and high-dimensional environments, where issues such as non-convexity, high dimensionality, and the presence of multiple local optima dominate the search landscape. In this context, recent years have seen a continuous rise in interest in metaheuristic methods, which offer flexible and stochastic search tools capable of addressing complex optimization problems without requiring derivative information or assumptions of continuity in the solution space.

Metaheuristic techniques are typically inspired by natural, biological, social, or physical processes, aiming to simulate powerful mechanisms for balancing exploration and exploitation within complex search spaces. Classic examples include Genetic Algorithms (GAs) [1], Particle Swarm Optimization (PSO) [2], and Ant Colony Optimization (ACO) [3], which have been widely used for decades. In recent years, however, a multitude of novel metaheuristics have emerged, motivated by the desire to overcome common limitations such as premature convergence and weak performance on rugged, multimodal landscapes [4]. These methods draw inspiration from a broad range of biological and ecological systems. From the animal kingdom, algorithms like Artificial Bee Colony (ABC) [5], Grey Wolf Optimizer (GWO) [6], the Whale Optimization Algorithm (WOA) [7], the Dragonfly Algorithm (DA) [8], Cuckoo Search (CS) [9], and the Bat Algorithm [10] emulate foraging, social, or navigation behaviors. Predator–prey-based algorithms such as Harris Hawks Optimization (HHO) [11] and the Snake Optimizer [12] capture hunting dynamics. Others derive from insect colonies or swarm intelligence, including the Firefly Algorithm [13], Glowworm Swarm Optimization (GSO) [14], and Butterfly Optimization [15]. Likewise, bacterial or microbial behaviors inspired algorithms such as Bacterial Foraging Optimization (BFO) [16], Virus Colony Search [17], and COVIDOA [18]. Some algorithms are motivated by botanical and plant behavior, such as the Plant Propagation Algorithm (PPA) [19], Invasive Weed Optimization (IWO) [20], and the Root Growth Optimizer [21]. Other methods are inspired by natural physical phenomena, including the Gravitational Search Algorithm (GSA) [22], Simulated Annealing [23], and the Harmony Search algorithm [24]. More recently, complex hybrid and bio-inspired models such as Gorilla Troops Optimization (GTO) [25], the Reptile Search Algorithm (RSA) [26], the Sine Cosine Algorithm (SCA) [27], and the Slime Mould Algorithm (SMA) [28] have been introduced. Despite the thematic diversity and creativity of modern bio-inspired metaheuristic techniques, many of them continue to face common shortcomings, such as the lack of truly adaptive dynamics, a static and rigid balance between exploration and exploitation, and the absence of documented convergence guarantees [29]. These limitations underline the need for next-generation algorithms capable of self-regulating their behavior according to the state of the search, maintaining stability in convergence, and faithfully reflecting the principles and rhythms of complex biological processes.

BioHealing Optimization (BHO) is positioned within this scientific and technological context, drawing inspiration from the regenerative process of wound healing in living organisms. Wound healing is a natural function characterized by a delicate balance between disruption and restoration, aimed at re-establishing homeostasis. BHO translates this biological principle into the optimization domain, creating a multi-phase methodology in which each phase has a distinct yet interdependent role:

• Injury phase: Rather than relying on static or simplistic modifications, BHO applies stochastic disturbances to selected dimensions of candidate solutions, emulating the initial, uncontrolled nature of biological injury. These disturbances may follow distributions that favor either gentler changes or rare, high-impact shifts, with their intensity dynamically adapted as the search progresses, encouraging broad exploration in the early stages and gradually reducing disruption near convergence.
• Healing phase: Subsequently, BHO selectively guides the modified dimensions toward the best-known solution in a process that mirrors the progressive restoration of biological tissues. This movement is neither mechanical nor fixed; the proportion and direction of adjustments are adapted to the search conditions, maintaining diversity while also enhancing the exploitation of high-quality solutions.
• Recombination phase: Optionally, this process is preceded by an information-exchange mechanism inspired by differential evolution (DE) [30,31,32], where components of the best solution are combined with differences from other members of the population. This allows the inheritance of strong traits while simultaneously introducing variations that keep the search active.

Relative to earlier healing-inspired optimizers such as Wound Healing Optimization (WHO) [33] and Synergistic Fibroblast Optimization (SFO) [34], BHO instantiates a concrete three-phase loop with per-dimension stochastic injury using Gaussian and Lévy steps under a decaying intensity schedule, guided healing toward the incumbent best, and an optional DE/best/1/bin recombination path. These operators are governed by stateful per-dimension controllers that update from accepted improvements, including a scar map with momentum, hot-dimension focusing, and stagnation-triggered RAGE and hyper-RAGE bursts, together with dynamic modulation of healing probabilities and intensities. As a result, BHO separates exploration from exploitation, preserves feasibility within bounds, and adapts both where it perturbs and how strongly it perturbs over time, providing capabilities that WHO and SFO do not explicitly implement.

Beyond healing-inspired methods, the last three years have produced a stream of contemporary population-based optimizers and refined DE and ES variants that frame the immediate state of the art for positioning BHO. Recent nature-inspired single-swarm methods include the Osprey Optimization Algorithm (OOA), the One-to-One-Based Optimizer (OOBO), the Secretary Bird Optimization Algorithm (SBOA), and the Dream Optimization Algorithm (DOA), each reporting competitive results on standard benchmarks and releasing enough detail for replication [35,36,37,38]. Within the differential evolution family, modern designs emphasize efficiency and adaptivity, for example, EBJADE with multi-population management and elite regeneration, iDE-APAMS with adaptive population allocation and mutation selection, and RLDMDE, which uses reinforcement learning to choose mutation strategies from diversity signals [39,40,41]. On the evolution strategies side for discrete and mixed variables, the line of work on (1 + 1)-CMA-ES with Margin and CMA-ES-SoP stabilizes sampling for integer-valued domains through margin control and point-set modeling [42,43]. Together, these threads capture the current landscape and provide up-to-date baselines against which BHO can be compared in the remainder of this paper.

Section 2 presents BHO. Section 2.1 provides the core pseudocode, and Section 2.2 details the adaptive mechanisms and their integration in the main loop. Section 3 describes the experimental protocol and the benchmark suite. Section 3.1 lists the test functions. Section 3.3 reports the parameter-sensitivity study. Section 3.3 presents the comparative experimental results under a fixed evaluation budget. Section 3.4 introduces the BHO-assisted neural network training case study on classification datasets. Section 4 states the conclusions, and Section 5 outlines directions for future research.

2. The BioHealing Optimization Algorithm

2.1. The Basic Body of the BioHealing Optimizer Pseudocode

The overall algorithm of the method is described below.

Notes for pseudocode:

$U(0,1)$ returns a sample from Uniform[0,1], randInt(1, $dim$) returns a uniformly random integer in {1, …, $dim$}. clamp(v,$lower$, u) projects v onto [$lower$,$upper$]. stochasticStep() produces a zero-mean random step; in practice, this is instantiated either as Gaussian $N(0,1)$ for fine local search or as a Lévy step (e.g., Mantegna) to enable rare large jumps; selection follows this paper’s described schedule/policy. healStep($h_r$) returns a scalar gain a used to bias coordinates toward $x_{best}$ when the per-dimension healing trigger $U(0,1)$ < $h_r$ fires. $FE$ counts every call to $f\left(·\right)$ (initialized after the first $NP$ evaluations and incremented after each new evaluation inside the loop). The $w_s$ schedule line maintains a small, nonzero intensity to avoid premature freezing late in the run.

The core loop of Algorithm 1 of the BHO maintains a population of candidate solutions within box constraints and repeatedly balances broad exploration with guided exploitation. It begins by sampling each vector uniformly within the per-dimension bounds, evaluating all candidates, and designating the incumbent best. At every iteration, the current elite is identified, and the wound intensity follows a monotone decay schedule so that early updates encourage wide exploration while later ones stabilize around promising regions. For each non-elite individual, an optional differential evolution recombination (best/1, bin) may combine the incumbent best with a scaled difference of two distinct peers; all values are kept feasible through clamping to the bounds. The injury phase then applies a per-dimension stochastic disturbance with a specified probability, using either Gaussian noise or a Lévy-tailed step produced by a generic stochasticStep() procedure and scaled by the current wound intensity and the variable range; feasibility is again enforced by clamping. The healing phase gently attracts modified components toward the incumbent best with a specified probability, using a step a = healStep($h_r$) that increases with the healing rate and preserves bounds. The resulting trial is evaluated and accepted greedily only if it improves the previous fitness; whenever an improvement is accepted, the global best is also updated. The procedure terminates upon exhausting either the iteration budget or the cap on function evaluations, and returns the pair (${\mathit{x}}_{best}$, $f_{best}$). This description captures the clean, modular backbone of BHO (elite selection, optional recombination, injury, healing, and greedy replacement) while allowing optional extensions to be integrated without altering the fundamental methodology.

Below are the core equations of BHO’s three components (injury, healing, and recombination), together with the minimal auxiliary relations required for completeness.

Notations:

• ${\mathbf{x}}_{i,j}^{\left(iter\right)}$: coordinate j of individual i at iteration $iter$;
• $[{\mathbf{lower}}_j,{\mathbf{upper}}_j]$: bounds;
• ${\mathbf{x}}_{best,j}^{\left(iter\right)}$: incumbent best;
• clamp(z, $\mathbf{lower}$, $\mathbf{upper}$) = min (max(z, $lower$), $\mathbf{upper}$);
• $U(0,1)$: uniform random in [0,1].

Algorithm 1 The basic body of the BioHealing Optimizer pseudocode

Input:    f: objective function to minimize    $dim$: problem dimensionality    $NP$: population size    $ite{r}_{max}$: maximum number of iterations    $FE$: counts all calls to $f\left(·\right)$ so far    $F{E}_{max}$: maximum number of function evaluations    $lower$, $upper$: bounds for each variable Params:    $w_{s0}$: initial wound intensity    $w_p$: probability of wounding per dimension    $h_r$: probability of healing per dimension    $r_p$: probability of recombination with the best solution    F : differential weight scaling factor in recombination    $CR$: crossover probability in recombination Output:    $x_{best}$: the best solution found    $f_{best}$: the value of f at best solution Initialization: 01 For i=1..$NP$ do: 02         For j=1..$dim$ do: 03                $x_{i,j}U(lowe{r}_j,uppe{r}_j)$ 04         End for 05 $fi{t}_i=f\left(x_i\right)$ 06 End for 07 $x_{best}$, $f_{best}$ = argmin($fi{t}_i$) // elite at initialization 08 $iter$ = 0 09 $FE$ = 0 Main loop: 10 While $iter$ < $ite{r}_{max}$ or $FE$ < $F{E}_{max}$ do 11         $iter$ = $iter$ + 1 12         $elite$ = argmin($fi{t}_i$) 13         $x_{best}$ = $x_{elite}$ 14         $f_{best}$ = $fi{t}_{elite}$ 15         // decay wound intensity but keep a safety floor at 5% of $w_{s0}$ 16         $w_s$ = max(0.05$·w_{s0}$, $w_{s0}·$(1 − ${\displaystyle \frac{iter}{ite{r}_{max}}}$)) 17         For i=1..$NP$ do 18                 If i = $elite$ then 19                         continue 20                 end if 21                 $x_{old}$ = $x_i$, $f_{old}$ = $fi{t}_i$ 22                 $f_{best}$ = $fi{t}_{elite}$ 23                  // Optional DE(best/1,bin) recombination 24                 if $U(0,1)$ < $r_p$ then 25                         Choose $r_1\ne i$, $r_2\ne i$, $r_2\ne r_1$ 26                         $j_r$ = randInt(1,$dim$) // ensure at least one dimension crosses 27                         For j=1..$dim$ do 28                               If $U(0,1)$ < $CR$ or j = $j_r$ then 29                                 v = $x_{best,j}$ + $F·$($x_{r1,j}$ − $x_{r2,j}$) 30                                 $x_{i,j}$= clamp(v, $lowe{r}_j$, $uppe{r}_j$) 31                              End if 32                         End for 33                End if 34                // Stochastic injury: Gaussian or Lévy step per dimension 35                For j=1..$dim$ do 36                        If $U(0,1)$ < $w_p$ then 37                              $\xi$ = stochasticStep() // returns zero-mean step, Gaussian or Lévy 38                              d = $w_s·$$\xi$$·$ ($uppe{r}_j$ − $lowe{r}_j$) 39                              $x_{i,j}$ = clamp($x_{i,j}$+ d, $lowe{r}_j$, $uppe{r}_j$) 40                        End if 41                End for 42                // Guided healing toward $x_{best}$ 43                a = healStep($h_r$) // scalar healing gain, e.g., in (0,1) 44                For j=1..$dim$ do 45                         If $U(0,1)$ < $h_r$ then 46                              $x_{i,j}$ = clamp($x_{i,j}$ + a($x_{bes{t}_j}$ − $x_{i,j}$), $lowe{r}_j$, $uppe{r}_j$) 47                         End if 48                End for 49                $f_{new}$ = f($x_i$) 50                $FE$ = $FE$ +1 51                // Greedy acceptance with rollback 52                If $f_{new}$ < $f_{old}$ then 53                         $fi{t}_i$ = $f_{new}$ 54                         If $f_{new}$ < $f_{best}$ then 55                              $f_{best}$ = $f_{new}$ 56                              $x_{best}$ = $x_i$ 57                        End if 58                Else 59                         $x_i$ = $x_{old}$ 60                         $fi{t}_i$ = $f_{old}$ 61                End if 62         End for 63 End while 64 Return $x_{best}$, $f_{best}$

1.

Injury (Stochastic perturbation)

Notations and definitions:

• Bounds and projection: lower and upper are vectors of per-dimension bounds, clamp (${\mathbf{lower}}_{\mathit{j}}$, ${\mathbf{upper}}_j$) projects a scalar to [${\mathbf{lower}}_{\mathit{j}}$, ${\mathbf{upper}}_j$];
• Wound intensity: $w_s^{\left(iter\right)}$ is the injury intensity at iteration t, $w_{s0}$ is its initial value.
• Stochastic step $\xi$: At iteration $iter$, ${\xi }_{i,j}^{\left(t\right)}$ is a zero-mean scalar, sampled independently across i, j, and $iter$ from either (i) a standard Gaussian (unit variance) or (ii) a Lévy/Mantegna generator with stability index $\alpha \in [1,2]$ and global-scale Lévy scale. An optional truncation ${\xi }_{i,j}^{\left(t\right)}\le {\xi }_{max}$ may be applied to avoid extreme outliers.

With per-dimension wound probability $w_p$ (or adaptive $w_{p,j}$):

  ${\mathbf{x}}_{i,j}^{\left(\mathrm{iter}\right)}\leftarrow clamp\left({\mathbf{x}}_{i,j}^{\left(\mathrm{iter}\right)}+w_s^{\left(\mathrm{iter}\right)}{\xi }_{i,j}^{\left(\mathrm{iter}\right)}({\mathbf{upper}}_{\mathit{j}}-{\mathbf{lower}}_{\mathit{j}}),\mathbf{lower},{\mathbf{upper}}_{\mathit{j}}\right)$

where ${\xi }_{i,j}^{\left(t\right)}=$ stochasticStep(), and the wound intensity $w_s^{\left(t\right)}$ can be scheduled, e.g.,

          $w_s^{\left(iter\right)}=max\left(0.05w_{s0},w_{s0}\left(1-\frac{iter}{ite{r}_{max}}\right)\right)$

Auxiliary (stochastic step definition):

    $\left\{\begin{array}{cc}\mathcal{N}(0,1),\hfill & \left(\mathrm{Gaussian}\right)\hfill \\ \mathrm{levyScale}{\displaystyle \frac{u}{{\left|v\right|}^{1/\alpha }}},u\sim \mathcal{N}(0,{\sigma }_u^2),v\sim \mathcal{N}(0,1),\hfill & (\mathrm{L}é\mathrm{vy}/\mathrm{Mantegna})\hfill \end{array}\right.$

with

            ${\sigma }_u={\left[{\displaystyle \frac{\Gamma (1+\alpha )sin\left(\pi \alpha /2\right)}{\Gamma \left(\frac{1+\alpha }{2}\right)\alpha 2^{(\alpha -1)/2}}}\right]}^{1/\alpha }$

and an optional global-scale Lévy scale.

The injury mechanism relies on stochastic perturbations that are complementary in their search behavior. Gaussian steps exhibit light tails and finite variance, which favor fine-grained local exploration and controlled diffusion around the current region of interest. In contrast, Lévy-distributed steps (with stability index $\alpha \in [1,2]$) have heavy tails, producing occasional long jumps that help the algorithm escape local basins and probe distant regions of the search space. By enabling per-dimension perturbations, BHO can adjust the disruption intensity across coordinates, which is beneficial in non-separable and multimodal landscapes where sensitivity is uneven across variables. This dual-mode design (Gaussian vs. Lévy) therefore provides a principled mechanism to balance local refinement and global exploration within the same injury phase.

2.

Healing (guided move toward ${\mathbf{x}}_{best}$)

Notations and definitions:

• Healing probability: $h_r\in [0,1]$ is the per-dimension probability that a healing move is applied (if adaptive per coordinate, write $h_{r,j}$).
• Healing gain: ${\alpha }^{\left(iter\right)}$ is a scalar returned by healStep($h_r$) at iteration iter; it controls the attraction strength toward the incumbent best. Unless stated otherwise, ${\alpha }^{\left(iter\right)}\in [0,1]$ and is non-decreasing in $h_r$ (e.g., the simple rule ${\alpha }^{\left(iter\right)}$ = 0.15 + 0.35 $h_r$, capped to stay below 1).
• Incumbent best: ${\mathbf{x}}_{best}^{iter}$ denotes the best solution available at iteration $iter$.
• Bounds and projection: lower and upper are vectors of per-dimension bounds; clamp (z, $\mathbf{lower}$, ${\mathbf{upper}}_j$) projects a scalar to [${\mathbf{lower}}_j$, ${\mathbf{upper}}_j$].
• Order of updates: Healing takes as input the post-injury state ${\mathbf{x}}_{i,j}^{(iter,\mathrm{inj})}$ and produces ${\mathbf{x}}_{i,j}^{(iter,\mathrm{heal})}$.

With per-dimension healing probability $h_r$:

   ${\mathbf{x}}_{i,j}^{(iter,\mathrm{heal})}=clamp\left({\mathbf{x}}_{i,j}^{(iter,\mathrm{inj})}+a^{\left(iter\right)}\left({\mathbf{x}}_{best,j}^{\left(iter\right)}-{\mathbf{x}}_{i,j}^{(iter,\mathrm{inj})}\right),{\mathbf{lower}}_j,{\mathbf{upper}}_j\right)$,

where $a^{\left(iter\right)}=\mathtt{healStep}\left(h_r\right)(\mathrm{e}.\mathrm{g}.,\mathrm{a}\mathrm{simple}\mathrm{linear}\mathrm{rule}{a}^{\left(iter\right)}=0.15+0.35h_r$, bounded in $[0,1)$)

3.

Recombination (DE/best/1/bin)

Notations and definitions:

• Recombination probability: $r_p$ is the probability that differential evolution (DE) recombination is applied to candidate i. Otherwise, the candidate proceeds without recombination.
• Mutation strategy: The variant used is DE/best/1, meaning one difference vector is added to the current best solution. Specifically, two distinct indices $r_1$, $r_2$≠i are sampled uniformly, and the donor vector is constructed as $\mathbf{v}={\mathbf{x}}_{best,j}^{\left(iter\right)}+F\left({\mathit{x}}_{r1,j}^{\left(iter\right)}-{\mathit{x}}_{r2,j}^{\left(iter\right)}\right)$.
• Scaling factor: $F\in [0,2]$ is the differential weight controlling the magnitude of the perturbation.
• Crossover mask: C is a binary mask with entries $C_j$∼Bernoulli($\mathrm{CR}$), where $CR\in [0,1]$ is the crossover probability. To ensure at least one coordinate crosses over, one randomly selected dimension $j_{rand}$ is forced to have $C_{j_{rand}}$= 1.
• Offspring construction: The trial vector after recombination is $x_i^{(iter,rec)}$, which combines donor and parent according to the mask and is then projected to the feasible range by clamp.

With probability $r_p$ the DE/best/1 with binomial crossover is applied:

            $\mathbf{v}=x_{best,j}^{\left(iter\right)}+F\left({\mathit{x}}_{r1,j}^{\left(iter\right)}-{\mathbf{x}}_{r2,j}^{\left(iter\right)}\right)$

          $\mathbf{C}\sim \mathrm{Bernoulli}\left(\mathrm{CR}\right),\mathrm{enforce}{\mathbf{C}}_{j_{\mathrm{rand}}}=1$

     ${\mathbf{x}}_{i,j}^{(iter,\mathrm{rec})}=clamp\left({\mathbf{C}}_j{\mathit{v}}_{i,j}^{\left(iter\right)}+(1-{\mathbf{C}}_J){\mathit{x}}_{i,j}^{\left(iter\right)},{\mathbf{lower}}_j,{\mathbf{upper}}_j\right)$

4.

Greedy acceptance

Notations and definitions:

• Trial fitness: After injury, healing, or recombination, the candidate ${\mathbf{x}}_i^{\left(iter·\right)}$ is evaluated to obtain $f_{new}$. The previous fitness is $f_{old}$.
• Greedy rule: The update is strictly elitist: if $f_{new}<f_{old}$, the candidate is accepted and its fitness is updated; otherwise, the previous state is restored.
• Best solution update: If the accepted candidate also improves upon the current best value $f_{best}$, then both $f_{best}$ and ${\mathit{x}}_{best}$ are updated.
• Rollback mechanism: If the trial vector fails to improve, the algorithm explicitly resets ${\mathbf{x}}_i\leftarrow {\mathbf{x}}_{old}$ and $f\leftarrow f_{old}$ to prevent quality degradation.

After the phases (in the algorithm’s prescribed order):

        $\mathrm{accept}{\mathbf{x}}_i^{\mathrm{new}}$$\mathrm{if}f\left({\mathbf{x}}_i^{\mathrm{new}}\right)<f\left({\mathbf{x}}_i^{\mathrm{old}}\right),$$\mathrm{else}\mathrm{revert}.$

2.2. Integration of Adaptive Mechanisms into the BHO Core Loop

The following mechanisms plug into the injury and healing stages of Algorithm 1, changing only how perturbations are generated and how guidance toward the best is applied, while selection and acceptance remain unchanged.

The integration of the extensions into the BHO core loop follows the method’s natural flow without altering its backbone. After initialization and before processing individuals in each iteration, the algorithm updates the stagnation status and configures any exploration bursts. At this point, a targeted micro-restart around the incumbent best may also be triggered when a lack of improvement persists. In the same pre-loop stage, the per-dimension importance scores are decayed, and the current hot set is selected so that subsequent perturbations are preferentially strengthened where recent gains have been observed.

At the heart of the iteration, immediately before applying stochastic disturbances, the noise generator is determined: the random step may use a heavy-tailed Mantegna draw to allow rare long jumps, or fall back to Gaussian noise, depending on the settings. The injury phase then exploits the hot-dimension priorities, the multiplicative burst effects when RAGE or hyper-RAGE windows are active, and a mild intensity boost when early stagnation is detected. The random step is gently biased toward the recent momentum direction of each coordinate through a dedicated bias coefficient, while bandage protection can be bypassed only during bursts so as not to throttle exploration. Immediately after the standard injuries, a targeted alpha-strike may rarely fire on a few coordinates, scaled by variable ranges and current wound strength, before control passes to healing.

The healing phase is modulated so that, during bursts, the restorative rate is temporarily reduced to avoid erasing exploration, followed by a short cooldown with amplified healing to smooth the return to stabilization. Optionally, while bursts are active, selected components may be copied from the incumbent best to inject direction without sacrificing diversity. Acceptance remains greedy, and only upon true improvement are the scar-map learning states updated: the per-dimension wounding probabilities and strengths, momentum, importance scores, and bandage. These updates feed the next injury cycle, conveying where and how stronger or more frequent disturbances are worthwhile. When no improvement occurs, the distance since the last best increases and can re-trigger soft boosts, bursts, or, if needed, micro-restarts at the beginning of the next iteration. In this way, the mechanisms are woven into injury, healing, and their transitions, while preserving the clean core architecture of elite selection, recombination, disturbance, restoration, and greedy replacement.

1.

Scar Map: Momentum and Bandage

Notations and definitions:

• Per-dimension wound schedule: ${\mathit{p}}^{\left(w\right)}=p_1^{\left(w\right)},\dots ,p_{dim}^{\left(w\right)}$, ${\mathit{s}}^{\left(w\right)}=s_1^{\left(w\right)},\dots ,s_{dim}^{\left(w\right)},$ clamped in [$p_{min}$, $p_{max}$] and [$s_{min}$, $s_{max}$].
• Momentum: $m_j\in$ [—1,1] = decayed running sign of recent accepted moves on dimension j.
• Improvement score: $d_j\ge$ 0 = per-dimension score.
• Bandage counter: $b_{i,j}\in$$N_0$ = freeze length for coordinate (i,j).
• Signals: $towardBes{t}_j\in$ [0,1] and $signDi{r}_j\in${−1,+1}.
• Hyper-parameters: learning rate $\eta$, momentum decay $\rho \in$(0,1), bias coefficient k > 0, bandage length $L_b$ with bounds $p_{min}$, $p_{max}$, $s_{min}$, $s_{max}$.

After greedy acceptance ($f_{new}$ < $f_{old}$), the scar map updates, per improved coordinate j, the wound probability $p_j^{\left(w\right)}$ and $s_j^{\left(w\right)}$ using learning rate $\eta$ and clamping to admissible ranges. The update strength grows with $towardBes{t}_j$, so dimensions aligned with the incumbent best are reinforced. Momentum is refreshed as

$$m_j\leftarrow (1-\rho )·m_j+\rho ·signDi{r}_j,$$

which biases future perturbations by $k{m}_j$. The score $d_j$ accumulates proportional improvement, supporting hot-dimension selection. Finally, the bandage sets $b_{i,j}\leftarrow L_b$, freezing recently improved coordinates.

Scar map is invoked immediately after acceptance; in subsequent iterations, injury samples per-dimension parameters $p_j^{\left(w\right)}$, $s_j^{\left(w\right)}$ (instead of a global $w_p$), applies bias from $m_j$, and skips coordinates with $b_{i,j}>$ 0. Healing remains unchanged and it graphically illustrate in Figure 1.

The algorithm for Scar map is provided in Algorithm 2.

Algorithm 2 Scar map: momentum and bandage

Input: Set D of improved coordinates, $towardBes{t}_j$, $signDi{r}_j$ for $j\in D$ Params: $\eta$, $p_{min}$, $p_{max}$, $s_{min}$, $s_{max}$, $\rho$, $\kappa$, $L_b$ State: $p_j^{\left(w\right)}$, $s_j^{\left(w\right)}$, $m_j$, $d_j,$$b_{i,j}$ 01 For j∈D do 02     $g_p$ = $\eta$$·$ (0.5 + 0.5 $·$ $towardBes{t}_j$) 03     $g_s$ = $\eta$$·$ (0.25 + 0.75 $·$ $towardBes{t}_j$) 04     $p_j^{\left(w\right)}$ = clamp($p_j^{\left(w\right)}$ + $g_p$, $p_{min}$, $p_{max}$) 05     $s_j^{\left(w\right)}$ = clamp($s_j^{\left(w\right)}$$g_p$, $p_{min}$, $p_{max}$)) 06     $m_j$ = (1 — $\rho$) $·$$m_j$ + $\rho$$·$$signDi{r}_j$ 07     $d_j$ = $d_j$ + improvement_signal() 08     $\mathrm{If}$$L_b$ > 0 then 09          $b_{i,j}$ = $L_b$ 10     End if 11     For each (i,j) do 12     If $b_{i,j}$ > 0 then 13         $b_{i,j}$= $b_{i,j}$ − 1 14         Continue 15     End if 15     Apply injury with prob. $p_j^{\left(w\right)}$, scale $s_j^{\left(w\right)}$, and bias $\kappa m_j$ If $mo{m}_{bia{s}_j}$ = $mo{m}_{bias}$$·$ sign($scarMomentu{m}_j$)

Scar map updates per-dimension memory and momentum after successful acceptance and provides the scores that hot-dims will use to focus exploration.

2.

Hot-Dims Focus (Top-K and Boosts for Injury)

Notations and definitions:

• Per-dimension score: $d_j\ge$ 0 accumulates recent accepted improvements (same $d_j$ used by scar map).
• Decay: $\delta \in$(0,1) applies a mild forgetting to down-weight stale gains.
• Top-K hot set: ${\mathcal{H}}_{iter}$ = $topK(d,K)$ (upright $\mathcal{H}$) selects the K highest-scoring dimensions at iteration $iter$.
• Boost factors (used at injury time only): ${\beta }_p\ge$ 1 for probability, ${\beta }_s\ge$ 1 for scale.
• These do not modify the stored $p_j^{\left(w\right)}$, $s_j^{\left(w\right)}$; they produce effective values for the current injury step:

  $${\tilde{p}}_j=\left\{\begin{array}{ll}min\left(p_{max},{\beta }_p{p}_j^{\left(w\right)}\right),\hfill & j\in {\mathcal{H}}_{iter},\hfill \\ p_j^{\left(w\right)},\hfill & \mathrm{otherwise},\hfill \end{array}\right.{\tilde{s}}_j=\left\{\begin{array}{cc}min\left(s_{max},{\beta }_s{s}_j^{\left(w\right)}\right),\hfill & j\in {\mathcal{H}}_{iter},\hfill \\ s_j^{\left(w\right)},\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$
  (Bounds $p_{min}$, $p_{max}$, $s_{min}$, $s_{max}$ as in scar map.)
• Integration: Hot-dims runs once per iteration before injury: decay $d_j$, select ${\mathcal{H}}_{iter}$, then injury uses ${\tilde{p}}_j$, ${\tilde{s}}_j$ in place of the base $p_j^{\left(w\right)}$, $s_j^{\left(w\right)}$. Scar-map updating of $d_j$ still happens after greedy acceptance. The hot dims algorithm is listed in Algorithm 3.

During injury (this iteration), use ${\tilde{p}}_j$ as the per-dimension injury probability and ${\tilde{s}}_j$ as the intensity scale in place of $p_j^{\left(w\right)}$, $s_j^{\left(w\right)}$; feasibility remains enforced via clamp. (No persistent change to the stored scar-map state.) The Hot-dims procedure is graphically illustrated in Figure 2.

Algorithm 3 Hot-dims focus: boosting probability and intensity on top-K dimensions

Params: $\delta$, K, ${\beta }_p$, ${\beta }_s$ State: $d_j$ (scores), $p_j^{\left(w\right)}$, $s_j^{\left(w\right)}$ from Scar Map 01 For j= 1..$dim$ do 02       $d_j$= (1 — $\delta$) $·$$d_j$ // mild forgetting 03 End for 04 ${\mathcal{H}}_{iter}$= $topK(d,K)$ 05 // Injury-time effective parameters 06 For j= 1..$dim$ do 07       If $j\in {\mathcal{H}}_{iter}$ then 08            ${\tilde{p}}_j$ = min($p_{max}$, ${\beta }_p·p_j^{\left(w\right)}$) 09            ${\tilde{s}}_j$ = min($s_{max}$), ${\beta }_s·s_j^{\left(w\right)}$) 10       Else 11            ${\tilde{p}}_j$ = $p_j^{\left(w\right)}$ 12            ${\tilde{s}}_j$ = $s_j^{\left(w\right)}$ 12       End if 13 End for

Once hot dimensions are identified, exploration intensity is dynamically escalated under stagnation through RAGE and hyper-RAGE.

3.

RAGE and Hyper-RAGE

Notations and definitions:

• Trigger threshold: $\tau >0$. A burst is armed when $\mathrm{imprSignal}\left(\right)\ge \tau$ and no cooldown is active.
• Burst length: $B\in {\mathbb{N}}_0$. During a burst, the counter $b^{\mathrm{rage}}>0$ is decremented each iteration.
• Hot-set magnifier: $\mu \ge 1$. While $b^{\mathrm{rage}}>0$, the hot set size is enlarged to $K^{\prime }=min(\dim ,\lceil \mu K\rceil )$ (otherwise K).
• Boost multipliers (RAGE): $\phi \ge 1$ for probability and $\gamma \ge 1$ for scale; these are applied $only$ during a burst.
• Score shaping (hyper-RAGE): $\nu \ge 0$ and per-dimension weights $w_j\in [0,1]$ obtained from the scores $d_j$ (e.g., $w_j=d_j/{max}_{\ell }{d}_{\ell }$ or rank-based).
• Cooldown: $L_c\in {\mathbb{N}}_0$. After a burst ends ($b^{\mathrm{rage}}=0$), a cooldown $c^{\mathrm{rage}}\leftarrow L_c$ prevents immediate retriggering.
• Effective injury parameters: As in hot-dims, the current-iteration values are ${\tilde{p}}_j,{\tilde{s}}_j$ (possibly boosted for $j\in {\mathcal{H}}_t$). The stored schedules $p_j^{\left(w\right)},s_j^{\left(w\right)}$ remain unchanged.

RAGE triggers immediately after greedy acceptance if $\mathrm{imprSignal}\left(\right)\ge \tau$ and $c^{\mathrm{rage}}=0$, opening a B-iteration window. Before injury, the hot set ${\mathcal{H}}_t$ is computed with size $K^{\prime }$ (if in burst) or K (otherwise). The injury step then uses the effective parameters

$${\widehat{p}}_j=min\left(p_{max},\phi {\tilde{p}}_j\right),{\widehat{s}}_j=min\left(s_{max},\gamma {\tilde{s}}_j\right),j\in {\mathcal{H}}_t,$$

and ${\widehat{p}}_j={\tilde{p}}_j$, ${\widehat{s}}_j={\tilde{s}}_j$ for $j\notin {\mathcal{H}}_t$.

Hyper-RAGE makes boosts dimension-aware via

$${\phi }_j=\phi \left(1+\nu w_j\right),{\gamma }_j=\gamma \left(1+\nu w_j\right),$$

so that higher-scoring coordinates receive proportionally stronger yet bounded pressure. When the burst ends, cooldown starts, and healing remains unchanged. All boosts are non-persistent (state schedules $p_j^{\left(w\right)},s_j^{\left(w\right)}$ are not overwritten). The rage procedure is given in Figure 3.

The Explosive exploration under stagnation procedure is described in Algorithm 4.

Algorithm 4 RAGE/hyper-RAGE: Explosive exploration under stagnation

Input current scores $d_j$, base per-dimension schedules $p_j^{\left(w\right)},s_j^{\left(w\right)}$, hot-set size K. Parameters: trigger $\tau >0$, burst length $B\in {\mathbb{N}}_0$, magnifier $\mu \ge 1$, boosts $\phi \ge 1$, $\gamma \ge 1$, shaping $\nu \ge 0$, cooldown $L_c\in {\mathbb{N}}_0$. State: burst counter $b^{\mathrm{rage}}\in {\mathbb{N}}_0$, cooldown $c^{\mathrm{rage}}\in {\mathbb{N}}_0$. 01 At acceptance (after $f_{\mathrm{new}}<f_{\mathrm{old}}$) 02 $\mathbf{If}$ $c^{\mathrm{rage}}=0$ $\mathbf{And}$ $\mathrm{imprSignal}\left(\right)=|f_{\mathrm{old}}-f_{\mathrm{new}}|\ge \tau$ $\mathbf{then}$ 03      $b^{\mathrm{rage}}=B$ 04 End if 05 //Before injury (each iteration): 06 Hot-set size: 07 $\mathbf{If}$ $b^{\mathrm{rage}}>0$ $\mathbf{then}$ 08      $K^{\prime }=min(\dim ,\lceil \mu K\rceil )$ 09 $\mathbf{Else}$ 10      $K^{\prime }=K$ 11 $\mathbf{End}\mathbf{if}$ 12 Hot set selection:${\mathcal{H}}_t\leftarrow \mathrm{topK}(d,K^{\prime })$. 13 Effective bases (from hot-dims): compute ${\tilde{p}}_j,{\tilde{s}}_j$ for all j (temporary, non-persistent). 14 //RAGE / hyper-RAGE boosts (for use in this injury): 15 For $j=1$,…,$\dim$$\mathbf{do}$ 16      If $b^{\mathrm{rage}}>0$$\mathbf{and}$ $j\in {\mathcal{H}}_t$$\mathbf{then}$ 17           // RAGE: uniform boosts on hot dimensions 18           $p_j=min\left(p_{max},\phi {\tilde{p}}_j\right)$, 19           $s_j=min\left(s_{max},\gamma {\tilde{s}}_j\right)$ 20           // Hyper-RAGE: score-shaped boosts (optional) 21           choose $w_j\in [0,1]$ from $d_j$ (e.g., $w_j=d_j/{max}_{\ell }{d}_{\ell }$) 22           ${\widehat{p}}_j=min\left(p_{max},\phi (1+\nu w_j){\tilde{p}}_j\right)$ 23           ${\widehat{s}}_j=min\left(s_{max},\gamma (1+\nu w_j){\tilde{s}}_j\right)$ 24      Else 25           $p={\tilde{p}}_j$ 26           $s={\tilde{s}}_j$ 27      End if 28 End For 29 //Pass to injury: use ${\widehat{p}}_j,{\widehat{s}}_j$ for the current iteration’s stochastic perturbations, feasibility via clamp remains unchanged. (Stored $p_j^{\left(w\right)},s_j^{\left(w\right)}$ are not overwritten.) 30 //After injury (end of iteration): 31 If $b^{\mathrm{rage}}>0$$\mathbf{then}$ 32      $b^{\mathrm{rage}}=b^{\mathrm{rage}}-1$, 33      If $b^{\mathrm{rage}}=0$$\mathbf{then}$ 34           $c^{\mathrm{rage}}=L_c$ 35      End if 35 Else 36      If $c^{\mathrm{rage}}>0$$\mathbf{then}$ 37      $c^{\mathrm{rage}}=c^{\mathrm{rage}}-1$ 38      End if 39 End if

Outside burst windows, the injury sampling is governed by Lévy-wounds to allow rare long jumps when progress diminishes.

4.

Lévy-Wounds (Mantegna)

Notations and definitions:

• Stability index: $\alpha \in (1,2]$.
• Global Lévy scale: $c_{\ell }>0$ (scalar).
• Per-dimension injury scheduling: use $p_j^{\left(w\right)}$ (probability) and $s_j^{\left(w\right)}$ (scale) from scar map/hot-dims; bandage/cooldowns apply as defined earlier.
• Stochastic step (Lévy/Mantegna) for iteration t and coordinate $(i,j)$,

  $$u\sim \mathcal{N}(0,{\sigma }_u^2),v\sim \mathcal{N}(0,1),$$

with

$${\sigma }_u={\left[{\displaystyle \frac{\mathsf{\Gamma }(1+\alpha )sin(\pi \alpha /2)}{\Gamma \left(\frac{1+\alpha }{2}\right)\alpha 2^{(\alpha -1)/2}}}\right]}^{1/\alpha }.$$

Optional truncation: $\left|{\xi }_{i,j}^{\left(t\right)}\right|\le {\xi }_{max}$ to avoid extreme outliers.

• Range scaling and projection: the raw move is scaled by $({\mathbf{upper}}_j-{\mathbf{lower}}_j)$ and projected with $\mathrm{clamp}\left(·\right)$ to enforce feasibility.

Lévy-wounds replaces the Gaussian injury step with a heavy-tailed Mantegna draw, producing infrequent long jumps that enhance basin escape while retaining many small moves. For each dimension j of candidate i, with probability $p_j^{\left(w\right)}$ we apply the update

$${\mathbf{x}}_{i,j}^{(t,\mathrm{inj})}\leftarrow \mathrm{clamp}\left({\mathbf{x}}_{i,j}^{\left(t\right)}+s_j^{\left(w\right)}{\xi }_{i,j}^{\left(t\right)}\left({\mathbf{upper}}_j-{\mathbf{lower}}_j\right),{\mathbf{lower}}_j,{\mathbf{upper}}_j\right).$$

The stability $\alpha$ controls tail/heaviness (smaller $\alpha$⇒ rarer but longer jumps), and $c_{\ell }$ sets the overall Lévy magnitude and can be tuned jointly with $s_j^{\left(w\right)}$. If momentum $m_j$ is active, a mild directional bias $\kappa m_j$ (with small $\kappa >0$) may be added to the step before clamping. Lévy-wounds integrates seamlessly with hot-dims and RAGE/hyper-RAGE: when a dimension j is hot, the effective injury parameters ${\tilde{p}}_j,{\tilde{s}}_j$ (or ${\widehat{p}}_j,{\widehat{s}}_j$ under bursts) are used in place of $p_j^{\left(w\right)},s_j^{\left(w\right)}$, while the Lévy draw ${\xi }_{i,j}^{\left(t\right)}$ remains unchanged. This procedure is given in Algorithm 5.

Algorithm 5 Lévy-wounds (Mantegna): heavy-tailed jumps for rare long-range moves

Input: $\alpha ,c_{\ell },p_j^{\left(w\right)},s_j^{\left(w\right)}$, optional $m_j,\kappa ,{\xi }_{max}$. State: candidate ${\mathbf{x}}_i$, bounds $\mathbf{lower},\mathbf{upper}$. 01 For $j=1\dots \dim$ do 02       If bandage is active on $(i,j)$ Then 03            Continue 04       End if 05       Draw $r\sim U(0,1)$ 06       If $r<p_j^{\left(w\right)}$ then 07            Sample $u\sim \mathcal{N}(0,{\sigma }_u^2)$, 08            $v\sim \mathcal{N}(0,1)$ 09            $\xi \leftarrow c_{\ell }u/{\left|v\right|}^{1/\alpha }$ 10            If truncation enabled then 11            $\xi \leftarrow \mathrm{clip}(\xi ,-{\xi }_{max},{\xi }_{max})$ 12            End if 13       (optional bias) $\xi \leftarrow \xi +\kappa m_j$ 14       $\Delta \leftarrow s_j^{\left(w\right)}\xi ({\mathbf{upper}}_j-{\mathbf{lower}}_j)$ 15       ${\mathbf{x}}_{i,j}^{(t,\mathrm{inj})}\leftarrow \mathrm{clamp}\left({\mathbf{x}}_{i,j}^{\left(t\right)}+\Delta ,{\mathbf{lower}}_j,{\mathbf{upper}}_j\right)$ 16       End if 17 End for

The flowchart of this method is given in Figure 4.

Beyond step distribution, alpha-strike adds rare targeted large moves on a few coordinates to vault past barriers.

5.

Alpha-Strike

Notations and definitions:

• Trigger condition: Alpha-strike activates when the algorithm stagnates for more than $T_{\mathrm{stall}}$ iterations (no improvement in $f_{\mathrm{best}}$) or when the diversity indicator D falls below a threshold $D_{min}$.
• Strike length: $L_{\alpha }\in \mathbb{N}$, the number of consecutive iterations in which strike mode remains active.
• Boost multipliers: ${\varphi }_{\alpha }\ge 1$ (probability) and ${\gamma }_{\alpha }\ge 1$ (scale), applied uniformly to hot dimensions during the strike.
• Hot set magnification: factor ${\mu }_{\alpha }\ge 1$, enlarging the set ${\mathcal{H}}_t$ selected from dimension scores.
• Reset option: fraction $\rho \in [0,1]$ of the worst individuals may be re-initialized uniformly in $[\mathbf{lower},\mathbf{upper}]$ to maximize disruption.

Mechanism (text). Alpha-strike is a deliberate, short burst of intensified perturbations designed to break strong stagnation. Once triggered, the algorithm enters a strike window of length $L_{\alpha }$. For all individuals and dimensions $j\in {\mathcal{H}}_t$, the effective injury parameters are magnified:

$${\widehat{p}}_j=min\left(p_{max},{\varphi }_{\alpha }{\tilde{p}}_j\right),{\widehat{s}}_j=min\left(s_{max},{\gamma }_{\alpha }{\tilde{s}}_j\right).$$

In addition, with probability $\rho$, entire individuals are reinitialized within the feasible bounds. When the strike window expires, normal operation resumes. Healing is not modified, while scar map, momentum, and dimension scores continue to update, ensuring smooth integration with the other auxiliary mechanisms. The Alpha-strike procedure is given in Algorithm 6.

Algorithm 6 Alpha-strike: targeted, rare, large jump on a few coordinates

Input:  $T_{\mathrm{stall}},D_{min},L_{\alpha },{\varphi }_{\alpha },{\gamma }_{\alpha },{\mu }_{\alpha },\rho$ State: best fitness $f_{\mathrm{best}}$, stagnation counter $t_{\mathrm{stall}}$, diversity indicator D, strike counter $b^{\alpha }$ 01 //At each iteration: 02 If $t_{\mathrm{stall}}>T_{\mathrm{stall}}$ or $D<D_{min}$ then 03      $b^{\alpha }=L_{\alpha }$ 04      $t_{\mathrm{stall}}=0$ 05 End if 06 If $b^{\alpha }>0$ then 07      Expand hot set: $K^{\prime }=min(\dim ,\lceil {\mu }_{\alpha }K\rceil )$ 08      ${\mathcal{H}}_t\leftarrow \mathrm{topK}(d,K^{\prime })$ 09      Compute effective parameters ${\tilde{p}}_j,{\tilde{s}}_j$ for all j 10      For each individual i do 11           Draw $r\sim U(0,1)$ 12           If $r<\rho$ then 13           Reinitialize ${\mathbf{x}}_i\sim U(\mathbf{lower},\mathbf{upper})$ 14           Continue to next individual 15           End if 16           For each $j\in {\mathcal{H}}_t$ do 17                ${\widehat{p}}_j\leftarrow min(p_{max},{\varphi }_{\alpha }{\tilde{p}}_j)$ 18                ${\widehat{s}}_j\leftarrow min(s_{max},{\gamma }_{\alpha }{\tilde{s}}_j)$ 19                Apply injury using ${\widehat{p}}_j,{\widehat{s}}_j$ 20           End for 21      End for 22      $b^{\alpha }\leftarrow b^{\alpha }-1$ 23 Else 24      Run normal injury / healing / recombination with parameters from scar map, hot-dims, RAGE/hyper-RAGE 25 End if

Notes.

• Alpha-strike is designed to be rare but disruptive, acting as a restart-like operator while preserving the algorithmic structure.
• Magnifiers ${\varphi }_{\alpha },{\gamma }_{\alpha }$ should be chosen conservatively (e.g., $2–3\times$ boosts) to avoid instability.
• The diversity indicator D can be population variance, centroid spread, or any normalized measure.

The flowchart of Alpha-strike is provided in Figure 5.

If mild or targeted boosts do not revive progress, a small-scale restart around the incumbent best follows.

6.

Catastrophic Micro-Reset

Notations and definitions:

• Trigger condition: Activates when stagnation persists despite alpha-strike and Lévy-wounds, i.e., after $T_{\mathrm{cat}}$ consecutive iterations without improvement in $f_{\mathrm{best}}$.
• Reset fraction: ${\rho }_{\mathrm{cat}}\in (0,1]$, fraction of the worst individuals to be replaced.
• Reset policy: New candidates are sampled uniformly within $[\mathbf{lower},\mathbf{upper}]$, optionally blended with the elite centroid.
• Cooldown: $C_{\mathrm{cat}}$ prevents consecutive activations within a short window.

Catastrophic micro-reset is a last-resort disruption operator. Instead of modifying coordinates dimension by dimension, it directly replaces a portion of the population with fresh samples, injecting diversity. It is “micro” because only a small fraction ${\rho }_{\mathrm{cat}}$ of individuals are reset, rather than the entire population, thus preserving useful structure and elite solutions. Optionally, reset positions may be drawn from a mixture of uniform distribution and a Gaussian centered on the elite centroid, balancing exploration of new areas and exploitation near promising regions. After activation, a cooldown $C_{\mathrm{cat}}$ ensures micro-reset cannot be triggered again too soon. The steps of the algorithm are shown in Algorithm 7.

Notes.

• Catastrophic micro-reset should be rare and disruptive, serving as a population-level restart.
• The fraction ${\rho }_{\mathrm{cat}}$ is usually small (e.g., 5–15%) to avoid destroying global progress.
• The cooldown $C_{\mathrm{cat}}$ prevents repeated resets and stabilizes the dynamics. The flowchart for this procedure is given in Figure 6.

  Algorithm 7 Catastrophic micro-reset: targeted mini-restart around the incumbent best

  Input: $T_{\mathrm{cat}},{\rho }_{\mathrm{cat}},C_{\mathrm{cat}}$ (optionally: elite centroid $\overline{\mathbf{x}}$, blend $\lambda \in [0,1]$) State: best fitness $f_{\mathrm{best}}$, stagnation counter $t_{\mathrm{stall}}$, cooldown counter $c_{\mathrm{cat}}$, population $x_{i=1}^{NP}$ with bounds $\mathbf{lower},\mathbf{upper}$ 01 At each iteration: 02 If $t_{\mathrm{stall}}>T_{\mathrm{cat}}$ and $c_{\mathrm{cat}}=0$ then 03      Identify worst set $\mathcal{W}$ of size $\lfloor {\rho }_{\mathrm{cat}}·NP\rfloor$ (largest objective values). 04      For each $i\in \mathcal{W}$ do 05           Sample $\mathbf{z}\sim U(\mathbf{lower},\mathbf{upper})$ 06           If elite biasing enabled then 07                 ${\mathbf{x}}_i=(1-\lambda )\mathbf{z}+\lambda \overline{\mathbf{x}}$ 08           Else 09                 ${\mathbf{x}}_i=\mathbf{z}$ 10           End if 11      End for 12      $t_{\mathrm{stall}}=0$ 13      $c_{\mathrm{cat}}=C_{\mathrm{cat}}$ 14 End if 15 If $c_{\mathrm{cat}}>0$ then 16      $c_{\mathrm{cat}}=c_{\mathrm{cat}}-1$ 17 End if 18 Continue normal injury/healing/recombination.

After a restart or a burst, a smooth return to stable exploitation is needed, which is controlled by healing adjustments with a short cooldown.

7.

Healing Adjustments and Cooldown

Notations and definitions:

• Adaptive healing gain: $a^{\left(t\right)}$, normally produced by healStep$\left(h_r\right)$, is scaled dynamically based on recent outcomes; cap $a_{max}$.
• Boost/decay factors: ${\beta }_h>1$ (boost on success), ${\delta }_h\in (0,1)$ (decay on failure).
• Cooldown threshold: ${\tau }_h\in (0,1)$; if a single healing move exceeds ${\tau }_h$ of the range, healing on that coordinate cools down.
• Cooldown length: $C_h\in \mathbb{N}$; per-dimension timer $c_j$.
• Counters: $s_j$ consecutive successes, $f_j$ consecutive failures (per dimension j).
• Bounds/projection: vectors $\mathbf{lower},\mathbf{upper}$; $\mathrm{clamp}(z,{\mathbf{lower}}_j,{\mathbf{upper}}_j)$.

After each healing attempt on coordinate j, the algorithm checks whether the new state improves fitness. On success, the healing gain is strengthened $a^{\left(t\right)}\leftarrow min(a_{max},{\beta }_h{a}^{\left(t\right)})$ and $s_j$ increments (reset $f_j=0$). On failure, the gain is reduced $a^{\left(t\right)}\leftarrow {\delta }_h{a}^{\left(t\right)}$ and $f_j$ increments (reset $s_j=0$). If the displacement magnitude exceeds a fraction ${\tau }_h$ of the variable’s range, a cooldown $c_j\leftarrow C_h$ starts, and while $c_j>0$, no healing is applied to that coordinate. This self-regulation complements scar map/bandage: bandage protects just-improved coordinates from injury, whereas cooldown throttles over-correction by healing. The steps of the algorithm are shown in Algorithm 8.

Algorithm 8 Healing adjustments and cooldown: modulating healing during bursts with a smooth post-burst ramp

Input:  ${\beta }_h,{\delta }_h,a_{max},{\tau }_h,C_h$ State: gains $a^{\left(t\right)}$, counters $s_j,f_j$, cooldown timers $c_j$, candidate ${\mathbf{x}}_i$, best ${\mathbf{x}}_{\mathrm{best}}$, bounds $\mathbf{lower},\mathbf{upper}$ 01 For each dimension j (on healing step) do 02      If $c_j>0$ then 03            $c_j=c_j-1$ 04            Continue 05      End if 06      Propose update: ${\mathbf{x}}_{i,j}^{(t,\mathrm{heal})}\leftarrow \mathrm{clamp}\left({\mathbf{x}}_{i,j}^{\left(t\right)}+a^{\left(t\right)}\left({\mathbf{x}}_{\mathrm{best},j}^{\left(t\right)}-{\mathbf{x}}_{i,j}^{\left(t\right)}\right),{\mathbf{lower}}_j,{\mathbf{upper}}_j\right).$ 07      //Evaluate fitness, let outcome be success if improved, else failure. 08      If success then 09            $s_j=s_j+1$ 10            $f_j=0$ 11            $a^{\left(t\right)}=min(a_{max},{\beta }_h{a}^{\left(t\right)})$ 12      Else 13            $f_j=f_j+1$; $s_j\leftarrow 0$ 14            $a^{\left(t\right)}={\delta }_h{a}^{\left(t\right)}$ 15      End if 16      IF $|{\mathbf{x}}_{i,j}^{(t,\mathrm{heal})}-{\mathbf{x}}_{i,j}^{\left(t\right)}|>{\tau }_h({\mathbf{upper}}_j-{\mathbf{lower}}_j)$ then 17            $c_j=C_h$ 18      End if 19 End for

Notes.

• Boosts/decays make healing stronger in promising directions and weaker in insensitive ones.
• Cooldown prevents oscillations and over-correction after large moves.
• Conservative tuning (e.g., ${\beta }_h\approx 1.1$, ${\delta }_h\approx 0.9$) is often adequate; $C_h$ small (e.g., 2–5). The flowchart for this method is shown in Figure 7.

For early or mild stagnation, a gentle injury amplification is applied before more aggressive mechanisms are engaged.

8.

Soft-Stagnation Boost

Notations and definitions:

• Global restart threshold: $T_{\mathrm{restart}}$, maximum stagnation length tolerated across all mechanisms.
• Restart fraction: ${\rho }_{\mathrm{restart}}\in (0,1]$, fraction of worst individuals replaced.
• Hybrid injection pool: ${\mathcal{P}}_{\mathrm{inj}}$, contains elite centroid, archived bests, or random samples.
• Blend parameter: $\lambda \in [0,1]$, probability of using ${\mathcal{P}}_{\mathrm{inj}}$ vs. uniform resampling.
• Diversity safeguard: $D_{min}$ must be re-established after restart; else ${\rho }_{\mathrm{restart}}$ increases temporarily.

Adaptive restarts and hybrid injection serve as a final safeguard against premature convergence. While scar map, hot-dims, RAGE/hyper-RAGE, alpha-strike, Lévy-wounds, healing and cooldown, and catastrophic micro-reset progressively increase disruption, there remains a chance that the population collapses to a narrow region. A global restart is triggered when no improvement in $f_{\mathrm{best}}$ occurs for more than $T_{\mathrm{restart}}$ iterations.

At restart, a fraction ${\rho }_{\mathrm{restart}}$ of the worst individuals is replaced. Each replacement is drawn using a hybrid scheme:

• With probability $\lambda$, sample from ${\mathcal{P}}_{\mathrm{inj}}$ (e.g., elite centroid, archived bests).
• With probability $1-\lambda$, sample uniformly from $[\mathbf{lower},\mathbf{upper}]$.

After injection, population diversity D is recomputed. If $D<D_{min}$, ${\rho }_{\mathrm{restart}}$ is temporarily increased until diversity is restored. This mechanism thus preserves elites while re-introducing fresh search directions. The steps for this method are shown in Algorithm 9.

Algorithm 9 Soft-stagnation boost: gentle injury amplification under early stagnation

Input:  $T_{\mathrm{restart}},{\rho }_{\mathrm{restart}},\lambda ,D_{min}$ State: stagnation counter $t_{\mathrm{stall}}$, population ${\mathbf{x}}_i$, injection pool ${\mathcal{P}}_{\mathrm{inj}}$, bounds $\mathbf{lower},\mathbf{upper}$ 01 At each iteration: 02 If $t_{\mathrm{stall}}>T_{\mathrm{restart}}$ then 03      Identify worst set $\mathcal{W}$ of size $\lfloor {\rho }_{\mathrm{restart}}·NP\rfloor$. 04      For each $i\in \mathcal{W}$ do 05            Draw $r\sim U(0,1)$ 06            If $r<\lambda$ then 07                 inject from ${\mathcal{P}}_{\mathrm{inj}}$ 08            Else 09                 0 ${\mathbf{x}}_i\sim U(\mathbf{lower},\mathbf{upper})$ 10            End if 11      End for 12      $t_{\mathrm{stall}}\leftarrow 0$ 13 End if 14 Recompute diversity D 15 If $D<D_{min}$ then 16      increase ${\rho }_{\mathrm{restart}}$ temporarily 17 End if 18 Continue normal injury/healing/recombination.

Notes.

• Acts as a population-level reboot, but partial: elites are preserved.
• ${\mathcal{P}}_{\mathrm{inj}}$ can evolve dynamically, e.g., storing historical elites.
• Parameters $T_{\mathrm{restart}}$ and ${\rho }_{\mathrm{restart}}$ should be tuned conservatively to avoid overly frequent resets. The flowchart for this method is shown in Figure 8.

  Figure 8. Soft-stagnation boost.

  Figure 8. Soft-stagnation boost.

These mechanisms integrate without altering the backbone of Algorithm 1 and are triggered under simple stagnation or improvement conditions. Also, the flowchart for this technique is shown in Figure 9.

3. Experimental Setup and Benchmark Results

This section first introduces the benchmark functions selected for experimental evaluation, followed by a comprehensive analysis of the experiments conducted. The study systematically examines the various parameters of the proposed algorithm to assess its reliability and effectiveness in different optimization scenarios. The experimental settings for the proposed method are outlined in

Table 1. Parameters and settings of BHO.

Group	Symbol/Name	Value	Description (Upright)
Core	$NP$	100	Population size.
	$ite{r}_{max}$	500	Max iterations (also used in $w_s$ schedule).
	$F{E}_{max}$	150,000	Max function evaluations (stopping).
	$w_{s0}$	0.14	Initial injury intensity; decays with $iter$.
	$w_p$	0.38	Per-dimension wounding probability.
	$h_r$	0.80	Per-dimension healing probability.
	$r_p$	0.15	Probability of DE (best/1,bin) move.
	F	0.70	DE differential weight (scale factor).
	$CR$	0.50	DE crossover probability.
	enabled	yes	Enable scar map.
	$\eta$	0.06	Learning rate for $p_j^{\left(w\right)},s_j^{\left(w\right)}$ updates.
	$L_b$	4	Bandage freeze length (iters).
	$p_{min}/p_{max}$	0.05/0.98	Bounds for per-dimension wound prob.
	$s_{min}/s_{max}$	0.50/3.00	Bounds for per-dimension wound scale.
	$\rho$	0.25	Momentum forgetting (decay).
	$\kappa$	0.75	Bias strength toward momentum direction.
Hot-Dims Focus	$\delta$	0.05	Decay rate for $d_j$ (dim scores).
	K	6	Number of top (hot) dimensions.
	${\beta }_p$	1.5	Prob. boost on hot-dims (effective, non-persistent).
	${\beta }_s$	1.6	Scale boost on hot-dims (effective, non-persistent).
RAGE/hyper-RAGE	enabled	yes	Enable RAGE bursts.
	$T_{\mathrm{rage}}$	12	No-improve threshold to arm RAGE (iters).
	B	10	RAGE burst length (iters).
	$\phi$	2.2	Prob. multiplier during RAGE.
	$\gamma$	2.8	Scale multiplier during RAGE.
	ignore bandage	yes	Ignore bandage during RAGE window.
	enabled	yes	Enable hyper-RAGE.
	$T_{\mathrm{HR}}$	28	Second stagnation threshold.
	$B_{\mathrm{HR}}$	12	Hyper-RAGE burst length (iters).
	${\phi }_{\mathrm{HR}}$	3.0	Prob. multiplier during hyper-RAGE.
	${\gamma }_{\mathrm{HR}}$	3.5	Scale multiplier during hyper-RAGE.
Lévy-Wounds (Mantegna)	enabled	yes	Enable Lévy steps.
	$\alpha$	1.5	Stability index (tail heaviness).
	$c_{\ell }$	0.6	Global Lévy scale.
Alpha-Strike	$\rho$	0.10	Reinit probability per individual within strike.
	${\sigma }_{\alpha }$	0.90	Strike step scaling modifier (implementation-specific).
Catastrophic Micro-Reset	enabled	yes	Enable micro-reset.
	$T_{\mathrm{cat}}$	40	No-improve iters ⇒ reset.
	${\rho }_{\mathrm{cat}}$	0.08	Fraction of population to reset.
	${\sigma }_{\mathrm{cat}}$	0.25	Std. around elite centroid for reset (if biased).
Healing Adjustments and Cooldown	$r_h^{\mathrm{burst}}$	0.35	Healing-rate reduction during bursts.
	$r_h^{\mathrm{post}}$	0.25	Healing-rate increase post-burst.
	$C_h$	8	Healing cooldown length (iters).
Adaptive Restarts and Hybrid Injection	$T_{\mathrm{restart}}$	10	Global stagnation threshold for restart.
	${\rho }_{\mathrm{restart}}$	—	Replacement fraction at restart (set per experiment).
	$\lambda$	—	Probability of injection from ${\mathcal{P}}_{\mathrm{inj}}$ (vs. uniform).
	$D_{min}$	—	Minimum target diversity after restart.

and for every other used method in

Table 2. Parameters and settings of other methods.

Name	Value	Description
$NP$	100	Population size for all methods
$ite{r}_{max}$	500	Maximum number of iterations for all methods
CLPSO
clProb	0.3	Comprehensive learning probability
cognitiveWeight	1.49445	Cognitive weight
inertiaWeight	0.729	Inertia weight
mutationRate	0.01	Mutation rate
socialWeight	1.49445	Social weight
CMA-ES
$N{P}_{CMA-ES}$	$4+⌊3·log\left(dim\right)⌋$	Population size
EA4Eig
archiveSize	100	Archive size for JADE-style mutation
eig_interval	5	Recompute eigenbasis every k iterations
maxCR	1	Upper bound for CR
maxF	1	Upper bound for F
minCR	0	Lower bound for CR
minF	0.1	Lower bound for F
pbest	0.2	p-best fraction (current-to-pbest/1)
tauCR	0.1	Self-adaptation prob. for CR
tauF	0.1	Self-adaptation prob. for F
mLSHADE_RL
archiveSize	500	Archive size
memorySize	10	Success-history memory size (H)
minPopulation	4	Minimum population size
pmax	0.2	Maximum p-best fraction
pmin	0.05	Minimum p-best fraction
SaDE
crSigma	0.1	Std for CR sampling
fGamma	0.1	Scale for Cauchy F sampling
initCR	0.5	Initial CR mean
initF	0.7	Initial F mean
learningPeriod	25	Iterations per adaptation window
UDE3
minPopulation	4	Minimum population size.
memorySize	10	Success-history memory size (H).
archiveSize	100	Archive size.
pmin	0.05	Minimum p-best fraction.
pmax	0.2	Maximum p-best fraction.

.

Beyond the settings summarized in the accompanying comparator table, we selected baselines that span the main metaheuristic families and design assumptions: CMA-ES for smooth, well-conditioned landscapes with strong local adaptation; DE variants (jDE, SaDE) for robust performance on multimodal or non-separable problems; advanced DE methods (mLSHADE_RL, UDE3) that maintain diversity and adapt step intensities; CLPSO, which often yields fast early descent but risks premature convergence; and EA4Eig, which leverages eigen-directions on anisotropic tasks. These choices follow methodological diversity, prevalence in the literature, code/settings availability, and reproducibility, ensuring a fair comparison under a harmonized evaluation budget and framing the analysis in Section 3.3.

3.1. Test Functions

The performance assessment of the proposed method was carried out using a comprehensive and diverse collection of well-established benchmark functions [44,45,46], as listed in

Table 3. The benchmark functions used in the experiments.

Problem	Formula	Dim	Bounds
Parameter Estimation for Frequency-Modulated Sound Waves	${min}_{x\in {[-6.4,6.35]}^6}f\left(x\right)={\displaystyle \frac{1}{N}}{\sum }_{n=1}^N{\left|y(n;x)-y_{\mathrm{target}}\left(n\right)\right|}^2$$y(n;x)=x_{0}sin\left(x_{1}n+x_{2}sin(x_{3}n+x_{4}sin\left(x_{5}n\right))\right)$	6	$x_i\in [-6.4,6.35]$
Lennard–Jones Potential	${min}_{x\in {\mathbb{R}}^{3N-6}}f\left(x\right)=4{\sum }_{i=1}^{N-1}{\sum }_{j=i+1}^N\left[{\left({\displaystyle \frac{1}{r_{ij}}}\right)}^{12}-{\left({\displaystyle \frac{1}{r_{ij}}}\right)}^6\right]$	30	$x_0\in (0,0,0)$ $x_1,x_2\in [0,4]$ $x_3\in [0,\pi ]$ $x_{3k-3}$ $x_{3k-2}$ $x_i\in [-b_k,+b_k]$
Bifunctional Catalyst Blend Optimal Control	${\displaystyle \frac{d{x}_1}{dt}}=-k_1{x}_1$, ${\displaystyle \frac{d{x}_2}{dt}}=k_1{x}_1-k_2{x}_2+k_3{x}_2+k_4{x}_3$, ${\displaystyle \frac{d{x}_3}{dt}}=k_2{x}_2$, ${\displaystyle \frac{d{x}_4}{dt}}=-k_4{x}_4+k_5{x}_5$, ${\displaystyle \frac{d{x}_5}{dt}}=-k_3{x}_2+k_6{x}_4-k_5{x}_5+k_7{x}_6+k_8{x}_7+k_9{x}_5+k_{10}{x}_7$ ${\displaystyle \frac{d{x}_6}{dt}}=k_8{x}_5-k_7{x}_6$, ${\displaystyle \frac{d{x}_7}{dt}}=k_9{x}_5-k_{10}{x}_7$ $k_i\left(u\right)=c_{i1}+c_{i2}u+c_{i3}{u}^2+c_{i4}{u}^3$	1	$u\in [0.6,0.9]$
Optimal Control of a Nonlinear Stirred- Tank Reactor	$J\left(u\right)={\int }_0^{0.72}\left[x_1{\left(t\right)}^2+x_2{\left(t\right)}^2+0.1u^2\right]dt$ ${\displaystyle \frac{d{x}_1}{dt}}=-2x_1+x_2+1.25u+0.5exp\left({\displaystyle \frac{x_1}{x_1+2}}\right)$ ${\displaystyle \frac{d{x}_2}{dt}}=-x_2+0.5exp\left({\displaystyle \frac{x_1}{x_1+2}}\right)$ $x_1\left(0\right)=0.9,x_2\left(0\right)=0.09$, $t\in [0,0.72]$	1	$u\in [0,5]$
Tersoff Potential for Model Si (B)	${min}_{\mathbf{x}\in \mathsf{\Omega }}f\left(\mathbf{x}\right)={\sum }_{i=1}^{N}E\left({\mathbf{x}}_i\right)$ $E\left({\mathbf{x}}_i\right)={\displaystyle \frac{1}{2}}{\sum }_{j\ne i}{f}_c\left(r_{ij}\right)\left[V_R\left(r_{ij}\right)-B_{ij}{V}_A\left(r_{ij}\right)\right]$ where $r_{ij}=\parallel {\mathbf{x}}_i-{\mathbf{x}}_j\parallel$, $V_R\left(r\right)=Aexp(-{\lambda }_{1}r)$ $V_A\left(r\right)=Bexp(-{\lambda }_{2}r)$ $f_c\left(r\right)$: cutoff function with $f_c\left(r\right)$: angle parameter	30	$x_1\in [0,4]$ $x_2\in [0,4]$ $x_3\in [0,\pi ]$ $x_i\in \left[{\displaystyle \frac{4(i-3)}{4}},4\right]$
Tersoff Potential for Model Si (C)	${min}_{\mathbf{x}}V\left(\mathbf{x}\right)={\sum }_{i=1}^N{\sum }_{j>i}^N{f}_C\left(r_{ij}\right)\left[a_{ij}{f}_R\left(r_{ij}\right)+b_{ij}{f}_A\left(r_{ij}\right)\right]$ $f_C\left(r\right)=\left\{\begin{array}{cc}1,\hfill & r<R-D\hfill \\ {\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{2}}cos\left({\displaystyle \frac{\pi (r-R+D)}{2D}}\right),\hfill & |r-R|\le D\hfill \\ 0,\hfill & r>R+D\hfill \end{array}\right.$ $f_R\left(r\right)=Aexp(-{\lambda }_{1}r)$ $f_A\left(r\right)=-Bexp(-{\lambda }_{2}r)$ $b_{ij}={\left[1+\left({\beta }^n\right){\zeta }_{ij}^n\right]}^{-1/\left(2n\right)}$ ${\sum }_{k\ne i,j}{f}_C\left(r_{ik}\right)g\left({\theta }_{ijk}\right)exp\left[{\lambda }_3^3{(r_{ij}-r_{ik})}^3\right]$	30	$x_1\in [0,4]$ $x_2\in [0,4]$ $x_3\in [0,\pi ]$ $x_i\in \left[{\displaystyle \frac{4(i-3)}{4}},4\right]$
Spread Spectrum Radar Polly phase Code Design	${min}_{x\in X}f\left(x\right)=max\left\{|{\phi }_1\left(x\right)|,|{\phi }_2\left(x\right)|,\dots ,|{\phi }_m\left(x\right)|\right\}$ $X=\{x\in {\mathbb{R}}^n\mid 0\le x_j\le 2\pi ,j=1,\dots ,n\}m=2n-1$ ${\phi }_j\left(x\right)=\left\{\begin{array}{cc}{\displaystyle \sum _{k=1}^{n-j}cos(x_k-x_{k+j})}\hfill & \mathrm{for}j=1,\dots ,n-1\hfill \\ {\displaystyle n}\hfill & \mathrm{for}j=n\hfill \\ {\displaystyle {\phi }_{2n-j}\left(x\right)}\hfill & \mathrm{for}j=n+1,\dots ,2n-1\hfill \end{array}\right.$ ${\phi }_j\left(x\right)={\sum }_{k=1}^{n-j}cos(x_k-x_{k+j}),j=1,\dots ,n-1$ ${\phi }_n\left(x\right)=n$, ${\phi }_{n+\ell }\left(x\right)={\phi }_{n-\ell }\left(x\right),\ell =1,\dots ,n-1$	20	$x_j\in [0,2\pi ]$
Transmission Network Expansion Planning	$min{\sum }_{l\in \mathsf{\Omega }}{c}_l{n}_l+W_1{\sum }_{l\in OL}|f_l-{\overline{f}}_l|+W_2{\sum }_{l\in \mathsf{\Omega }}max(0,n_l-{\overline{n}}_l)$ $Sf=g-d$ $f_l={\gamma }_l{n}_l\Delta {\theta }_l,\forall l\in \mathsf{\Omega }$ $|f_l|\le {\overline{f}}_l{n}_l,\forall l\in \mathsf{\Omega }$ $0\le n_l\le {\overline{n}}_l,n_l\in \mathbb{Z},\forall l\in \mathsf{\Omega }$	7	$0\le n_i\le {\overline{n}}_l$ $n_i\in \mathbb{Z}$
Electricity Transmission Pricing	${min}_{x}f\left(x\right)={\sum }_{i=1}^{N_g}{\left({\displaystyle \frac{C_i^{gen}}{P_i^{gen}}}-R_i^{gen}\right)}^2+{\sum }_{j=1}^{N_d}{\left({\displaystyle \frac{C_j^{load}}{P_j^{load}}}-R_j^{load}\right)}^2$ ${\sum }_{j}G{D}_{i,j}+{\sum }_{j}B{T}_{i,j}=P_i^{gen},\forall i$ ${\sum }_{i}G{D}_{i,j}+{\sum }_{i}B{T}_{i,j}=P_j^{load},\forall j$ $G{D}_{i,j}^{max}=min(P_i^{gen}-B{T}_{i,j},P_j^{load}-B{T}_{i,j})$	126	$G{D}_{i,j}\in [0,G{D}_{i,j}^{max}]$
Circular Antenna Array Design	${min}_{r_1,\dots ,r_6,{\phi }_1,\dots ,{\phi }_6}f\left(\mathbf{x}\right)={max}_{\theta \in \mathsf{\Omega }}AF(\mathbf{x},\theta )$ $AF(\mathbf{x},\theta )=\left|{\sum }_{k=1}^{6}exp\left(j\left[2\pi r_{k}cos(\theta -{\theta }_k)+{\phi }_k{\displaystyle \frac{\pi }{180}}\right]\right)\right|$	12	$r_k\in [0.2,1]$ ${\phi }_k\in [-180,180]$
Dynamic Economic Dispatch 1	${min}_{\mathbf{P}}f\left(\mathbf{P}\right)={\sum }_{t=1}^{24}{\sum }_{i=1}^5\left(a_i{P}_{i,t}^2+b_i{P}_{i,t}+c_i\right)$ $P_i^{min}\le P_{i,t}\le P_i^{max},\forall i=1,\dots ,5,t=1,\dots ,24$ ${\sum }_{i=1}^5{P}_{i,t}=D_t,\forall t=1,\dots ,24$ $P_{min}=[10,20,30,40,50]$ $P_{max}=[75,125,175,250,300]$	120	$P_i^{min}\le P_{i,t}\le P_i^{max}$
Dynamic Economic Dispatch 2	${min}_{\mathbf{P}}f\left(\mathbf{P}\right)={\sum }_{t=1}^{24}{\sum }_{i=1}^9\left(a_i{P}_{i,t}^2+b_i{P}_{i,t}+c_i\right)$ $P_i^{min}\le P_{i,t}\le P_i^{max},\forall i=1,\dots ,5,t=1,\dots ,24$ ${\sum }_{i=1}^5{P}_{i,t}=D_t,\forall t=1,\dots ,24$ $P_{min}=[150,135,73,60,73,57,20,47,20]$ $P_{max}=[470,460,340,300,243,160,130,120,80]$	216	$P_i^{min}\le P_{i,t}\le P_i^{max}$
Static Economic Load Dispatch (1, 2, 3, 4, 5)	${min}_{P_1,\dots ,P_{N_G}}F={\sum }_{i=1}^{N_G}{f}_i\left(P_i\right)$ $f_i\left(P_i\right)=a_i{P}_i^2+b_i{P}_i+c_i,i=1,2,\dots ,N_G$ $f_i\left(P_i\right)=a_i{P}_i^2+b_i{P}_i+c_i+|e_{i}sin\left(f_i(P_i^{min}-P_i)\right)|$ $P_i^{min}\le P_i\le P_i^{max},i=1,2,\dots ,N_G$ ${\sum }_{i=1}^{N_G}{P}_i=P_D+P_L$ $P_L={\sum }_{i=1}^{N_G}{\sum }_{j=1}^{N_G}{P}_i{B}_{ij}{P}_j+{\sum }_{i=1}^{N_G}{B}_{0i}{P}_i+B_{00}$ $P_i-P_i^0\le U{R}_i{P}_i^0-P_i\le D{R}_i$	6 13 15 40 140	See Technical Report of CEC2011

. These test functions represent a standard suite commonly utilized in the global optimization literature for validating and comparing metaheuristic algorithms. Each function exhibits distinct characteristics in terms of modality, separability, dimensionality, and landscape complexity, thus providing a robust basis for evaluating the generalization capability of the algorithm. Notably, the functions were employed in their original, unaltered form; no additional transformations such as shifting, rotation, or scaling were applied, allowing for a transparent and reproducible comparison with prior studies.

Table 4. Properties of benchmark functions used.

Problem	Modality	Separability	Conditioning
Parameter Estimation for Frequency-Modulated Sound Waves	Multimodal	Non-separable	Moderate
Lennard–Jones Potential	Near-flat/ effectively	Non-separable	Ill-conditioned
Bifunctional- Catalyst Blend Optimal Control	Near-flat / effectively unimodal	Non-separable	Near-flat/ well-conditioned
Optimal Control of a Nonlinear Stirred- Tank Reactor	Near-flat/ effectively unimodal	Non-separable	Near-flat/ well-conditioned
Tersoff Potential for Model Si (B)	Multimodal	Non-separable	Ill-conditioned
Tersoff Potential for Model Si (C)	Multimodal	Non-separable	Ill-conditioned
Spread-Spectrum Radar Polly Phase- Code Design	Multimodal	Non-separable	Moderate
Transmission Network Expansion Planning	Near-flat/ effectively unimodal	Non-separable	Near-flat
Electricity Transmission Pricing	Structured/ likely multimodal	Non-separable	Moderate
Circular Antenna Array Design	Multimodal	Non-separable	Moderate
Dynamic Economic Dispatch 1	Smooth (nearly quadratic)	Non-separable
Dynamic Economic Dispatch 2	Structured/ possibly multimodal	Non-separable	Moderate
Static Economic Load Dispatch 1	Multimodal (valve-point effects)	Non-separable	Ill-conditioned
Static Economic Load Dispatch 2	Multimodal (valve-point effects)	Non-separable	Ill-conditioned
Static Economic Load Dispatch 3	Multimodal (valve-point effects)	Non-separable	Ill-conditioned
Static Economic Load Dispatch 4	Multimodal (valve-point effects)	Non-separable	Ill-conditioned
Static Economic Load Dispatch 5	Multimodal (valve-point effects)	Non-separable	Ill-conditioned

provides a standardized description of the test set along three dimensions: modality (unimodal vs. multimodal), separability (separable vs. non-separable), and conditioning (well-/ill-conditioned or near-flat where applicable). The labels reflect the structural forms presented in the problem definitions (Table 3) and the qualitative behavior noted in the text for near-flat cases. This overview is intended to document the landscape characteristics of the suite without anticipating performance conclusions.

3.2. Parameter Sensitivity

Following the parameter-sensitivity protocol introduced by Lee et al. [28], we provide a systematic foundation for analyzing algorithmic responsiveness to configuration changes and the extent to which reliability is preserved across diverse operating conditions. The results are shown in

Table 5. Sensitivity analysis of the method parameters for the “Parameter Estimation for Frequency-Modulated Sound Waves” problem.

Parameter	Value	Mean	Min	Max	Iters	Main-Effect Range
$wp$	0.1	0.138292	$2.55618\times {10}^{20}$	0.274259	150	0.0066789
	0.3	0.141141	$3.30715\times {10}^{-17}$	0.274394	150
	0.5	0.135853	$8.66494\times {10}^{21}$	0.234322	150
	0.7	0.136441	$6.84876\times {10}^{-20}$	0.272128	150
	0.9	0.134732	$3.07218\times {10}^{-21}$	0.21554	150
$hr$	0.1	0.133427	$8.66494\times {10}^{-21}$	0.221196	150	0.00544895
	0.3	0.138797	$5.62092\times {10}^{-17}$	0.274394	150
	0.5	0.137095	$3.07218\times {10}^{-21}$	0.274259	150
	0.7	0.137095	$5.95873\times {10}^{-20}$	0.235609	150
	0.9	0.138533	$2.08349\times {10}^{-16}$	0.23928	150

.

The graphical representation of $wp$ and $hr$ for the “Parameter Estimation for Frequency-Modulated Sound Waves” problem is shown in Figure 10.

A sensitivity analysis for the Lennard–Jones problem is shown in

Table 6. Sensitivity analysis of the method parameters for the “Lennard–Jones Potential” problem.

Parameter	Value	Mean	Min	Max	Iters	Main-Effect Range
$wp$	0.1	−27.1628	−32.3234	−19.509	150	0.247722
	0.3	−27.2453	−32.3291	−20.1075	150
	0.5	−27.1499	−33.3141	−18.626	150
	0.7	−26.9975	−33.0304	−20.6726	150
	0.9	−27.243	−33.216	−19.6771	150
$hr$	0.1	−27.0514	−33.0304	−20.1075	150	0.565867
	0.3	−27.1281	−32.3291	−18.626	150
	0.5	−27.5176	−33.3141	−21.4231	150
	0.7	−27.1498	−33.216	−19.509	150
	0.9	−26.9517	−31.699	−19.6771	150

.

Also, a graphical representation of $wp$ and $hr$ for the “Lennard–Jones Potential” problem is shown in Figure 11.

A sensitivity analysis for for the “Tersoff Potential for Model Si (B)” problem is given in

Table 7. Sensitivity analysis of the method parameters for the “Tersoff Potential for Model Si (B)” problem.

Parameter	Value	Mean	Min	Max	Iters	Main-Effect Range
$wp$	0.1	−26.2187	−27.4563	−24.7514	150	0.20291
	0.3	−26.2884	−28.385	−24.788	150
	0.5	−26.1057	−27.5849	−25.209	150
	0.7	−26.3086	−28.1733	−24.8844	150
	0.9	−26.2761	−27.4406	−25.5369	150
$hr$	0.1	−26.3295	−28.385	−24.7514	150	0.164339
	0.3	−26.2855	−28.1733	−25.1714	150
	0.5	−26.1948	−27.7785	−24.788	150
	0.7	−26.2224	−27.1772	−25.2564	150
	0.9	−26.1652	−27.5849	−25.1464	150

.

The sensitivity study for wp and hr was conducted on three representative problems with N = 150 runs per setting and is summarized via the mean best indicator, complemented by min/max extremes. The main-effect range, defined as the spread of mean best across the scanned values, serves as a comparable measure of parameter influence. Also, a graphical representation of $wp$ and $hr$ for the “Tersoff Potential for Model Si (B)” problem is shown in Figure 12.

For Parameter Estimation of FM Sound Waves, the main-effect ranges are very small for both parameters ($w_p$: 0.0067, $h_r$: 0.00545), i.e., roughly 4–5% of the typical mean best level (∼0.137). This pattern indicates strong robustness to moderate mis-specification. Slight improvements are observed toward higher wp (≈0.9) and lower $h_r$ (≈0.1), but the gains are marginal and do not justify aggressive fine-tuning. The extremes (Min on the order of ${10}^{-20}$ and Max around 0.27) remain stable across settings, reflecting the spread expected from repeated independent runs under a fixed budget.

For the Lennard–Jones potential, sensitivity is more pronounced and asymmetric across parameters: the main effect is moderate for $w_p$ (0.248) and clearly larger for $h_r$ (0.566). Because lower (more negative) values are better here, the mean performance improves around intermediate hr (≈0.5), while moving toward the extremes degrades the average. The $w_p$ effect is milder but present, with a mild preference for mid-to-higher values. The best observed minima (down to —33.31) appear near intermediate settings, which is consistent with the need to balance injury intensity and healing rate on rugged, non-separable landscapes.

For Tersoff (Si, B) the picture is intermediate: main effects are modest ($w_p$: 0.203, $h_r$: 0.164) relative to the mean best level (∼—26.2), corresponding to sub-percent relative changes. Mean performance tends to favor slightly lower hr (≈0.1–0.3), while wp performs well in the mid-to-high range (≈0.7). The smallest minima are obtained for $w_p$≈ 0.3 and $h_r$≈ 0.1, suggesting that occasional deep improvements are more reliably triggered in these zones, even though the mean shows limited variation.

Taken together, the results substantiate that the algorithm is robust to reasonable parameter deviations, with negligible sensitivity on FM Sound Waves, a stronger dependence on hr for Lennard–Jones, and mild-to-moderate sensitivity on Tersoff (Si, B). A practical guideline is that a single conservative default can be used across the suite, while targeted, problem-aware adjustments are beneficial for highly multimodal and poorly conditioned cases, where hr in particular appears to govern the exploration–exploitation balance most effectively. While min/max broaden our sense of dispersion, conclusions should primarily rely on the mean behavior under the prescribed multi-run protocol.

3.3. Experimental Results

All experimental procedures were executed on a high-performance computer equipped with a machine running Debian Linux with 16 cores. The evaluation protocol was designed to ensure statistical rigor and reproducibility. Specifically, each benchmark function was subjected to 30 independent runs, with each trial initialized using distinct random seeds to account for stochastic variability in the algorithm’s behavior.

The proposed method and all comparators were implemented in C++ language and integrated into the open-source OPTIMUS framework [47]. The source code is publicly available at https://github.com/itsoulos/GLOBALOPTIMUS (accessed on 9 September 2025) to promote transparency and reproducibility. The environment was Debian 12.12, GCC 13.4. The primary performance metric is the average number of objective function evaluations over 30 runs for each test function. We also report success rates (in parentheses next to the mean), defined as the percentage of runs that reached the global optimum, when all runs converged optimally, and this indicator is omitted for clarity. In the results tables, best values are shown in bold and second-best values are underlined.

With a fixed termination of 500 iterations and all other parameters as in Table 1, we measured the wall-clock time (s) for the proposed method on two classic tests as dimensionality (D) increased from 20 to 200 (Figure 13). The results indicate an almost linear growth with dimension, consistent with the per-iteration complexity O($NP·dimension$): for Ellipsoidal, the average slope was about 0.139 s per additional dimension (≈2.78 s per +20D), whereas for Rosenbrock, it was ≈0.149 s per dimension (≈2.97 s per +20D). Both curves increased monotonically, while Rosenbrock exhibited a slightly steeper slope and a noticeable uptick beyond 160D, which was consistent with the higher per-evaluation cost induced by its cross-dimensional coupling. A crossover was observed: at low dimensions, Rosenbrock ran faster than Ellipsoidal, but from ~160D onward, Rosenbrock became slower, indicating that the interaction terms started to dominate the arithmetic in high D. Overall, these findings confirm the expected near-linear scaling of BHO with dimension under a fixed iteration budget, with function-dependent differences driven primarily by the cost of evaluating $f\left(·\right)$.

The test functions were as follows:

• Ellipsoidal: $f\left(x\right)={\sum }_{i=1}^n{\left({10}^6\right)}^{{\displaystyle \frac{i-1}{n-1}}}{x}_i^2$;
• Rosenbrock: $f\left(x\right)={\sum }_{i=1}^{n-1}\left(100{\left(x_{i+1}-x_i^2\right)}^2+{\left(x_i-1\right)}^2\right),-30\le x_i\le 30$.

For the experimental evaluation of BHO, we selected the following optimization algorithms as baselines for comparison:

• EA4Eig [35], a cooperative evolutionary algorithm with an eigen-crossover operator. It was introduced at the IEEE Congress on Evolutionary Computation 2022 (CEC 2022) and evaluated on that event’s official single-objective benchmark suite. The method was primarily designed and tested within the CEC single-objective context.
• UDE3 (UDE-III) [36], a recent member of the Unified DE family for constrained problems, it was evaluated on the CEC 2024 constrained set, while its predecessor (UDE-II/IUDE) won first place at CEC 2018, giving UDE3 a clear lineage with proven competitive performance.
• mLSHADE_RL [37] is a multi-operator descendant of LSHADE-cnEpSin, and was one of the CEC 2017 winners for real-parameter optimization. This variant augments the base algorithm with restarts and local search, and it has been assessed on modern test suites.
• CLPSO [38], a classic PSO variant with comprehensive learning (2006) that has long served as a strong baseline in CEC benchmarks; it is not tied to a single award entry but is widely used in comparative studies.
• SaDE [39], the self-adaptive DE developed by Qin and Suganthan, was evaluated on the CEC 2005 test set and has since become a reference point for adaptive strategies.
• jDE [40], developed by Brest et al., introduced self-adaptation of F and CR and has been extensively evaluated, including special CEC 2009 sessions on dynamic/uncertain optimization. This work helped popularize self-adaptation in DE.
• CMA-ES [48] is the established covariance matrix adaptation evolution strategy for continuous domains. Beyond its vast literature, it is a staple baseline and frequent participant/reference in BBOB benchmarking at GECCO, effectively serving as a standard black-box comparison standard.

On synthetic and physical potentials, BHO delivered particularly strong best values—nearly zero error on Parameter Estimation for Frequency-Modulated Sound Waves—and attained the lowest best value on the Lennard–Jones Potential. However, for the mean on the Lennard–Jones Potential, jDE ranked first, followed by CMA-ES, indicating that classical, Gaussian-driven strategies remain very stable when the landscape exhibits symmetric curvature. On the Tersoff Potential for Model Si (B) and Tersoff Potential for Model Si (C), the picture shifted: for Si (B), the best mean was achieved by EA4Eig, with CMA-ES close behind. While UDE3 achieved the best best, for Si (C), EA4Eig achieved the best best and CMA-ES achieved the best mean. Overall, advanced DE variants with stronger recombination (EA4Eig, UDE3) and CMA-ES alternated at the top, which aligns with the literature on rough but moderately structured landscapes.

In electro-economic and industrial test cases, the pattern diverged in informative ways. On Electricity Transmission Pricing, BHO attained the lowest best, whereas UDE3 achieved the best mean, suggesting that BHO can reach excellent extremes, while UDE3 maintained consistently low variability across runs. In Dynamic Economic Dispatch 1 and Dynamic Economic Dispatch 2, CMA-ES dominated first place in both the best and mean, confirming its strength on smooth, nearly quadratic geometries. In second place, BHO recorded the lowest mean and EA4Eig the best best, indicating that BHO’s injury-healing dynamics act as a variance-damping safety net. Across the Static Dispatch problems, the results were mixed: in Static Economic Load Dispatch 1, the best best was achieved by mLSHADE_RL, and the best mean was achieved by EA4Eig; in Static Economic Load Dispatch 2, mLSHADE_RL yielded the lowest mean (with EA4Eig achieving the best best); and in Static Economic Load Dispatch 4, mLSHADE_RL again achieved the best best, while EA4Eig achieved the best mean. Static Economic Load Dispatch 3 showed practical ties around the same value and was not very discriminative. Static Economic Load Dispatch 15 contained a notable outlier, where BHO’s mean was orders of magnitude lower than the others. This result merits independent verification, such as re-running experiments or checking units, because the unusually large gap likely signals a scaling difference or an unexpected effect. A comparison of algorithms based on best and mean after $1.5e\times {10}^5$ FEs is shown in

Table 8. Comparison of algorithms based on best and mean after $1.5e\times {10}^5$ FEs.

Function	EA4Eig Best/Mean (2022) [35]	UDE3 Best/Mean (2024) [36]	mLSHADE_RL Best/Mean (2024) [37]	CLPSO Best/Mean (2006) [38]	SaDE Best/Mean (2009) [39]	jDE Best/Mean (2006) [40]	CMA-ES Best/Mean (2001) [48]	BHO Best/Mean
Parameter Estimation for Frequency-Modulated Sound Waves	0.153993305	0.03808755	0.116157535	0.1314837477	0.095829444	0.116157541	0.18160916	$1.543774\times {10}^{-25}$
	0.213447296	0.115833815	0.205011062	0.2124981688	0.195600253	0.146008756	0.256863966	0.200220764
Lennard–Jones Potential	−18.48174236	−21.41786661	−28.41816707	−13.43649135	−21.93636189	−29.98126575	−28.42253189	−32.07742417
	−16.3133561	−17.33977959	−22.49792055	−10.25073403	−17.95333019	−27.49258505	−25.78783328	−24.33212306
Bifunctional- Catalyst Blend Optimal Control	−0.000286591	−0.000286591	−0.000286591	−0.000286591	−0.000286591	−0.000286591	−0.000286591	−0.000286591
	−0.000286591	−0.000286591	−0.000286591	−0.000286591	−0.000286591	−0.000286591	−0.000286591	−0.000286591
Optimal Control of a Nonlinear Stirred-Tank Reactor	0.390376723	0.390376723	0.390376723	0.3903767228	0.390376723	0.390376723	0.3903767228	0.3903767228
	0.390376723	0.390376723	0.390376723	0.3903767228	0.390376723	0.390376723	0.390376723	0.390376723
Tersoff Potential for Model Si (B)	−29.11960284	−29.44152761	−28.60814558	−28.23544117	−27.25703406	−13.51157064	−29.26244222	−29.03183049
	−27.89597789	−25.70627602	−26.07976794	−26.18834522	−25.25867422	−3.983690794	−27.5889735	−27.26630121
Tersoff Potential for Model Si (C)	−33.39767521	−33.12997729	−32.28575942	−30.85200257	−31.85343594	−18.76214649	−33.19699356	−33.38947338
	−31.11610936	−28.6603137	−30.03594436	−28.87349048	−29.59692733	−8.506037168	−31.79270914	−31.31864105
Spread-Spectrum Radar Polly Phase-Code Design	0.517866993	1.048240196	0.033146096	1.085334991	0.572731322	1.525870558	0.01484822722	0.195096433
	0.838752978	1.265152951	0.625788451	1.343956153	0.844014075	1.812042166	0.171988666	0.601190863
Transmission Network Expansion Planning	250	250	250	250	250	250	250	250
	250	250	250	250	250	250	250	250
Electricity Transmission Pricing	13,773,680	13773582.53	13773567.36	13775010.1	13773468.68	13774627.84	13775841.77	13773334.9
	13774198.28	13773582.53	13773852.63	13775395.07	13773930.93	14020953.78	13787550.18	13773632.45
Circular Antenna Array Design	0.006809638	0.006809653	0.006809701	0.006933401045	0.00681287	0.006820072	0.007204797576	0.007101505
	0.006809638	0.011722944	0.006823547	0.05181551798	0.008186204	0.017657998	0.008635655364	0.158523549
Dynamic Economic Dispatch 1	412736103.9	410197836.9	415275891.6	428607927.6	411226317.3	968042312.1	88285.6024	410074526.4
	421199260.5	410628483.3	418526775.3	435250914.5	413699347.4	1034393036	102776.7103	410079513.3
Dynamic Economic Dispatch 2	346855.5418	357530.5408	392247.7213	33031590.31	519820.5596	340091475.3	502699.4187	347469.862
	12332507.25	537163.4139	5966956.365	53906147.38	4090304.18	397471715.1	477720.1511	354734.7105
Static Economic Load Dispatch 1	6163.560978	6164.766919	6163.546883	6554.672173	6360.353305	6163.749006	6657.613028	6512.525519
	6170.965013	6266.250251	6353.722019	7668.333603	6464.861828	6778.527028	415917.4625	6772.880257
Static Economic Load Dispatch 2	14.46160489	18725.64707	17905.85383	19030.36081	18455.37286	1161578.904	763001.2185	18754.99866
	18779.92036	19660.49074	18661.20763	20699.00219	21829.10208	3671587.605	1425815.44	19232.05211
Static Economic Load Dispatch 3	470023233.3	470023232.3	470023232.6	470192288.3	470023232.7	471058115.8	470023232.3	470023233.2
	470023234.5	470023232.3	470023234.7	470294703.2	470023232.7	471963142.3	470023232.3	470023278
Static Economic Load Dispatch 4	71193.07649	168334.7003	71067.8441	884980.5569	862196.432	6482592.714	476053.5197	71089.03508
	71193.07649	348720.6677	406986.2181	1423887.358	1469886.142	17527314.24	2925852.935	100831.805
Static Economic Load Dispatch 5	8085796774	8070408727	8079118012	8105947615	8078489742	8453090778	8072077963	8072061692
	8145304780	8071802922	8104455647	8110924071	8081680478	8459337082	8084017791	74458095

.

On more “classic” comparative tests, CMA-ES showed impressive robustness on smooth, well-scaled landscapes: beyond Dynamic Economic Dispatch 1, it also achieved both the best best and mean on Spread-Spectrum Radar Polly Phase-Code Design. EA4Eig stood out when strong anisotropy or correlation mattered. Circular Antenna Array Design was surpassed by EA4Eig in both the best best and mean, consistent with its eigen-crossover design. UDE3 performed very well on problems with additional structure or constraints, such as the mean on Electricity Transmission Pricing, matching its remit as a unified DE for constrained scenarios. mLSHADE_RL frequently attained best-case wins in Static Economic Load Dispatch 1 and Static Economic Load Dispatch 4, and remained competitive in the mean across several cases, underscoring the value of ensemble mutation plus restarts on fractured landscapes. SaDE, while a solid baseline, tended to trail the newer DE descendants, and CLPSO underperformed in most tables, as expected where anisotropy demanded directional information beyond standard swarm-velocity updates. jDE remained competitive on certain physical potentials, for example, the mean on Lennard–Jones, but showed larger dispersion on industrial dispatch tasks.

On flat or near-flat landscapes (Bifunctional-Catalyst Blend Optimal Control, Transmission Network Expansion Planning, and essentially Optimal Control of a Nonlinear Stirred-Tank Reactor), where differences were on the order of ${10}^{-10}$, all methods tied or were practically indistinguishable, so these tests add little diagnostic power. In sum, there was no single winner: CMA-ES was the reference choice for smooth, well-conditioned cases and remained very strong on mean performance; EA4Eig excelled where alignment with principal variance directions helped; UDE3 often won on mean under constrained or pricing-structure scenarios; mLSHADE_RL frequently achieved best-case wins on difficult static dispatch variants; and BHO achieved the top best values on several critical functions and competitive means in demanding settings, indicating an effective exploration-to-exploitation transition.

Linking outcomes to the benchmark landscape (see Table 4) revealed consistent patterns. On multimodal, non-separable problems such as FM Sound Waves, Lennard–Jones Potential, and Electricity Transmission Pricing, BHO frequently attained first-place “best” values, whereas CMA-ES was strongest on smooth, well-conditioned cases (e.g., Dynamic Economic Dispatch 1). EA4Eig and mLSHADE_RL traded advantages on anisotropic or fractured landscapes (e.g., Tersoff and Static Economic Load Dispatch). By contrast, three near-flat/structured tests (Bifunctional-Catalyst Blend Optimal Control, Transmission Network Expansion Planning, and Stirred-Tank Reactor) were effectively non-discriminative and had little impact on the global ordering. The rank-based view in

Table 9. Detailed ranking of algorithms based on best after $1.5\times {10}^5$ FEs.

Function	EA4Eig	UDE3	mLSHADE_RL	CLPSO	SaDE	jDE	CMA-ES	BHO
	(2022) [35]	(2024) [36]	(2024) [37]	(2006) [38]	(2009) [39]	(2006) [40]	(2001) [48]
Parameter Estimation for Frequency-Modulated Sound Waves	7	2	4	6	3	5	8	1
Lennard–Jones Potential	7	6	4	8	5	2	3	1
Bifunctional- Catalyst Blend Optimal Control	1	1	1	1	1	1	1	1
Optimal Control of a Nonlinear Stirred- Tank Reactor	4	4	4	1	4	4	1	1
Tersoff Potential for Model Si (B)	3	1	5	6	7	8	2	4
Tersoff Potential for Model Si (C)	1	4	5	7	6	8	3	2
Spread-Spectrum Radar Polly Phase- Code Design	4	6	2	7	5	8	1	3
Transmission Network Expansion Planning	1	1	1	1	1	1	1	1
Electricity Transmission Pricing	5	4	3	7	2	6	8	1
Circular Antenna Array Design	1	2	3	6	4	5	8	7
Dynamic Economic Dispatch 1	5	3	6	7	4	8	1	2
Dynamic Economic Dispatch 2	1	3	4	7	6	8	5	2
Static Economic Load Dispatch 1	2	4	1	7	5	3	8	6
Static Economic Load Dispatch 2	1	4	2	6	3	8	7	5
Static Economic Load Dispatch 3	1	1	4	7	5	8	1	6
Static Economic Load Dispatch 4	3	4	1	7	6	8	5	2
Static Economic Load Dispatch 5	6	1	5	7	4	8	3	2
Total	53	51	55	98	71	99	66	47

and

Table 10. Detailed ranking of algorithms based on mean after $1.5\times {10}^5$ FEs.

Function	EA4Eig	UDE3	mLSHADE_RL	CLPSO	SaDE	jDE	CMA-ES	BHO
	(2022) [35]	(2024) [36]	(2024) [37]	(2006) [38]	(2009) [39]	(2006) [40]	(2001) [48]
Parameter Estimation for Frequency-Modulated Sound Waves	7	1	5	6	3	2	8	4
Lennard–Jones Potential	7	6	4	8	5	1	2	3
Bifunctional- Catalyst Blend Optimal Control	1	1	1	1	1	1	1	1
Optimal Control of a Nonlinear Stirred- Tank Reactor	2	2	2	1	2	2	2	2
Tersoff Potential for Model Si (B)	1	6	5	4	7	8	2	3
Tersoff Potential for Model Si (C)	3	7	4	6	5	8	1	2
Spread-Spectrum Radar Polly Phase- Code Design	4	6	3	7	5	8	1	2
Transmission Network Expansion Planning	1	1	1	1	1	1	1	1
Electricity Transmission Pricing	5	1	3	6	4	8	7	2
Circular Antenna Array Design	1	5	2	7	3	6	4	8
Dynamic Economic Dispatch 1	6	3	5	7	4	8	1	2
Dynamic Economic Dispatch 2	6	3	5	7	4	8	2	1
Static Economic Load Dispatch 1	1	2	3	7	4	6	8	5
Static Economic Load Dispatch 2	2	4	1	5	6	8	7	3
Static Economic Load Dispatch 3	4	1	5	7	3	8	1	6
Static Economic Load Dispatch 4	1	3	4	5	6	8	7	2
Static Economic Load Dispatch 5	7	2	5	6	3	8	4	1
Total	59	54	58	91	66	99	59	48

separates peak performance (“best”) from reliability (“mean”) under a fixed $1.5\times {10}^5$-FE budget, reflecting these patterns.

The evaluation distinguished peak performance (“best”) from reliability (“mean”) over seventeen problems. Summing per-function ranks showed that BHO had the lowest totals in both views (best: 47; mean: 48) and therefore the best overall score (overall: 95; average rank: 2.794). UDE3 followed closely (best/mean: 51/54; overall: 105; average rank: 3.088). EA4Eig and mLSHADE_RL formed the next tier with near-equal totals (112 and 113; average ranks: 3.294 and 3.323). CMA-ES sat mid-pack (overall: 125; average rank: 3.676), followed by SaDE (overall: 137; average rank: 4.029), while CLPSO and jDE ranked lowest overall (overall: 189 and 198; average ranks: 5.558 and 5.823). These rankings reflect consistently higher placements across most functions.

The detailed “best” table shows that BHO frequently ranked first in demanding tasks (e.g., Parameter Estimation for Frequency-Modulated Sound Waves, Lennard–Jones Potential, and Electricity Transmission Pricing), indicating strong exploration and high upside. UDE3 recorded many top-three finishes and several wins, especially where structure/constraints were prominent, explaining its overall second place and good generalization. EA4Eig and mLSHADE_RL traded advantages on anisotropic or fractured landscapes (e.g., Tersoff Si(B)/Si(C) and Static Economic Load Dispatch), consistent with eigen-guided recombination in EA4Eig and ensemble mutation with restarts in mLSHADE_RL. CMA-ES showed the expected resilience on smooth, well-conditioned geometries; its mean rank was often better than its best rank, reflecting low variance rather than aggressive extremes. SaDE remained a sturdy baseline but lagged behind newer DE descendants. CLPSO underperformed (as is typical under strong anisotropy). jDE exhibited occasional peaks but greater dispersion, which inflated its ranks in both the best and mean.

A comparison of all algorithms and a final ranking is given in

Table 11. Comparison of algorithms and final ranking.

Algorithm	Best	Mean	Overall	Average	Rank
BHO	47	48	95	2.794	1
UDE3 (2024) [36]	51	54	105	3.088	2
EA4Eig (2022) [35]	53	59	112	3.294	3
mLSHADE_RL (2024) [37]	55	58	113	3.323	4
CMA-ES (2001) [48]	66	59	125	3.676	5
SaDE (2009) [39]	71	66	137	4.029	6
CLPSO (2006) [38]	98	91	189	5.558	7
jDE (2006) [40]	99	99	198	5.823	8

.

Contrasting the best and mean revealed performance consistency. BHO’s lead in best values did not come at the expense of reliability; its mean total was also the lowest, indicating that top outcomes were not isolated “lucky” runs. Likewise, UDE3’s small best-mean gap indicates stable performance across repeats. EA4Eig and mLSHADE_RL displayed complementary behaviors (alignment with principal directions for EA4Eig, collective mutations and restarts for mLSHADE_RL). CMA-ES often “won” on mean where smoothness enforced small fluctuations.

Three problems were practically non-discriminative. In Bifunctional-Catalyst Blend Optimal Control and Transmission Network Expansion Planning, all methods tied, and in Optimal Control of a Nonlinear Stirred-Tank Reactor, differences were negligible. These problems diluted separability without altering the final ordering. By contrast, problems such as Electricity Transmission Pricing and the Dynamic/Static Economic Load Dispatch families yielded substantive differences, driving a clear lead for BHO and UDE3, and a tight contest for third and fourth place between EA4Eig and mLSHADE_RL.

In summary, with a fixed budget of $1.5·{10}^5$ evaluations, BHO was the strongest overall method in both peak and average performance, followed closely by UDE3 with high consistency, then tEA4Eig and mLSHADE_RL, which leveraged different mechanisms. CMA-ES remained a reliable reference baseline, while SaDE, CLPSO, and jDE underperformed under the present conditions.

3.4. BHO-Assisted Neural Network Training

We conducted an additional experiment in which BHO was used to train artificial neural networks [49,50] by minimizing the training error defined as

$$E\left(N\left(\overrightarrow{x},\overrightarrow{w}\right)\right)=\sum _{i=1}^M{\left(N\left({\overrightarrow{x}}_i,\overrightarrow{w}\right)-y_i\right)}^2$$

(2)

In this expression, $N\left(\overrightarrow{x},\overrightarrow{w}\right)$ denotes the neural network applied to an input vector $\overrightarrow{x}$, and $\overrightarrow{w}$ is the network’s parameter vector. The set $\left(\overrightarrow{x_i},y_i\right),i=1,\dots$$,M$ is the training dataset, with $y_i$ representing the desired targets for each pattern $\overrightarrow{x_i}$. To assess BHO across diverse conditions, we employed a broad set of classification datasets obtained from public repositories:

• The UCI database: https://archive.ics.uci.edu/ (accessed on 5 July 2025) [51].
• The Keel website: https://sci2s.ugr.es/keel/datasets.php (accessed on 5 July 2025) [52].
• The Statlib website: https://lib.stat.cmu.edu/datasets/index (accessed on 5 July 2025).

The experiments were conducted using the following datasets:

• APPENDICITIS, a medical dataset [53].
• ALCOHOL, a dataset related to alcohol consumption [54].
• AUSTRALIAN, a dataset derived from various bank transactions [55].
• BALANCE [56], a dataset derived from various psychological experiments.
• CLEVELAND, a medical dataset that was discussed in a series of papers [57,58].
• CIRCULAR, an artificial dataset.
• DERMATOLOGY, a medical dataset for dermatology problems [59].
• ECOLI, a dataset related to protein problems [60].
• GLASS, a dataset containing measurements from glass component analysis.
• HABERMAN, a medical dataset related to breast cancer.
• HAYES-ROTH [61].
• HEART, a dataset related to heart diseases [62].
• HEARTATTACK, a medical dataset for the detection of heart diseases.
• HOUSEVOTES, a dataset related to Congressional voting in the USA [63].
• IONOSPHERE, a dataset containing measurements from the ionosphere [64,65].
• LIVERDISORDER, a medical dataset extensively studied in a series of papers [66,67].
• LYMOGRAPHY [68].
• MAMMOGRAPHIC, a medical dataset used for the prediction of breast cancer [69].
• PARKINSONS, a medical dataset used for the detection of Parkinson’s disease [70,71].
• PIMA, a medical dataset for the detection of diabetes [72].
• PHONEME, a dataset that contains sound measurements.
• POPFAILURES, a dataset related to experiments regarding climate [73].
• REGIONS2, a medical dataset applied to liver problems [74].
• SAHEART, a medical dataset concerning heart diseases [75].
• SEGMENT [76].
• STATHEART, a medical dataset related to heart diseases.
• SPIRAL, an artificial dataset with two classes.
• STUDENT, a dataset related experiments in schools [77].
• TRANSFUSION, which is a medical dataset [78].
• WDBC, a medical dataset related to breast cancer [79,80].
• WINE, a dataset related to measurements of the quality of wines [81,82].
• EEG, a dataset containing EEG recordings [83,84]. From this dataset the following cases were used: Z_F_S, ZO_NF_S, ZONF_S, and Z_O_N_F_S.
• ZOO, a dataset related to animal classification [85].
  Interpreting the entries as classification error rates, where lower is better, BHO attained the lowest average error at 21.52%, ahead of Genetic at 26.55%, BFGS at 34.34%, and Adam at 34.48%. This corresponded to a 5.03 percentage-point improvement over Genetic (≈19% relative reduction) and ≈ a 13 percentage-point improvement over Adam and BFGS (≈37% relative reduction). At the dataset level, BHO recorded the best error on a majority of datasets, including ALCOHOL 16.89%, AUSTRALIAN 29.20%, BALANCE 8.06%, CLEVELAND 46.36%, CIRCULAR 4.23%, DERMATOLOGY 9.89%, ECOLI 48.72%, HEART 20.96%, HEARTATTACK 21.06%, HOUSEVOTES 6.45%, LYMOGRAPHY 27.24%, PARKINSONS 14.72%, PIMA 29.40%, REGIONS2 26.83%, SAHEART 33.88%, SEGMENT 27.26%, SPIRAL 43.75%, STATHEART 20.71%, TRANSFUSION 24.56%, WDBC 6.43%, WINE 15.86%, ZO_NF_S 9.76%, ZONF_S 2.68%, and ZOO 7.27%, and it tied for best on MAMMOGRAPHIC at 17.24%. There were, however, datasets where other optimizers had an edge, notably Adam on APPENDICITIS and STUDENT, and Genetic on GLASS, IONOSPHERE, LIVERDISORDER, and Z_F_S, with BHO typically close behind in these cases. Overall, the pattern showed that BHO delivered consistently low error across heterogeneous benchmarks, while conceding a few instances where gradient-based or classical evolutionary baselines aligned better with the data geometry. Reporting dispersion measures or statistical tests alongside these point estimates would further substantiate the observed gains.

Significance levels were computed in R on the classification error data and encoded, as shown in Figure 14, with ns for p > 0.05, * for p < 0.05, ** for p < 0.01, *** for p < 0.001, and **** for p < 0.0001. For the pairwise comparisons, BHO vs. Adam, BHO vs. BFGS, and BHO vs. Genetic, we obtained p = ****, i.e., p < 0.0001, indicating extremely significant differences. Combined with BHO’s lower average error rates, this suggests that its advantage is not due to random variation but reflects a systematic performance difference.

4. Conclusions

We presented BioHealing Optimization, a population-based metaheuristic that blends stochastic “injury,” guided “healing” toward the incumbent best, and an optional DE(best/1,bin) recombination step, augmented by adaptive per-dimension controllers (scar map and momentum, hot-dimension focusing, RAGE/hyper-RAGE bursts, Lévy steps, and healing modulation). The architecture self-regulates exploration–exploitation while preserving the core loop of elite selection, recombination, disturbance, greedy acceptance, and restoration.

Our protocol used 30 independent runs per problem, a fixed budget of $1.5·{10}^5$ evaluations, and harmonized parameter disclosure. Rank aggregation over 17 problems separated peak performance (best-of-runs) from reliability (mean-of-runs). Within this suite and budget, BHO attained the best combined ranking (best/mean), while strong baselines such as UDE3, EA4Eig, mLSHADE_RL, and CMA-ES remained competitive. This does not imply dominance on every instance, but supports the view that injury–healing–recombination with adaptive, per-dimension control is an effective strategy for challenging continuous optimization.

We commonly observed rapid early descent from Lévy/Gaussian perturbations, plateau escape when RAGE/hyper-RAGE fired, and stabilization via healing/protection (scar/bandage). On smooth, well-conditioned landscapes, CMA-ES often enjoyed faster early convergence, whereas BHO tended to excel on multimodal or non-separable cases.

The findings hold under the fixed budget and the evaluated corpus. Performance depends on problem traits (smoothness, separability, conditioning), and heavy-tailed steps may need conservative tuning under tight constraints. Mechanism overhead is linear in D and modest relative to evaluation cost, though non-zero. Sensitivity to key parameters (e.g., $w_p$, $h_r$) is small to moderate, not null. Non-discriminative tests (near-ties) do not drive the conclusions. Adding a real-world case (e.g., Lennard–Jones) strengthens practical evidence but does not exhaust application breadth.

Additionally, Section 3.4 assessed external validity by applying BHO to the training of feedforward neural networks on thirty-three public classification datasets. Under the same protocol and evaluation budget, BHO attained the lowest mean classification error (21.52 percent), compared with Genetic (26.55 percent), BFGS (34.34 percent), and Adam (34.48 percent), as summarized in

Table 12. Experimental results on the classification datasets across various machine learning methods. The bold notation is used for the method with the lowest error.

Dataset	Adam	BFGS	Genetic	BHO
APPENDICITIS	16.50%	18.00%	24.40%	22.07%
ALCOHOL	57.78%	41.50%	39.57%	16.89%
AUSTRALIAN	35.65%	38.13%	32.21%	29.20%
BALANCE	12.27%	8.64%	8.97%	8.06%
CLEVELAND	67.55%	77.55%	51.60%	46.36%
CIRCULAR	19.95%	6.08%	5.99%	4.23%
DERMATOLOGY	26.14%	52.92%	30.58%	9.89%
ECOLI	64.43%	69.52%	54.67%	48.72%
GLASS	61.38%	54.67%	52.86%	53.43%
HABERMAN	29.00%	29.34%	28.66%	28.82%
HAYES-ROTH	59.70%	37.33%	56.18%	35.49%
HEART	38.53%	39.44%	28.34%	20.96%
HEARTATTACK	45.55%	46.67%	29.03%	21.06%
HOUSEVOTES	7.48%	7.13%	6.62%	6.45%
IONOSPHERE	16.64%	15.29%	15.14%	15.54%
LIVERDISORDER	41.53%	42.59%	31.11%	31.87%
LYMOGRAPHY	39.79%	35.43%	28.42%	27.24%
MAMMOGRAPHIC	46.25%	17.24%	19.88%	17.24%
PARKINSONS	24.06%	27.58%	18.05%	14.72%
PIMA	34.85%	35.59%	32.19%	29.40%
POPFAILURES	5.18%	5.24%	5.94%	7.40%
REGIONS2	29.85%	36.28%	29.39%	26.83%
SAHEART	34.04%	37.48%	34.86%	33.88%
SEGMENT	49.75%	68.97%	57.72%	27.26%
SPIRAL	47.67%	47.99%	48.66%	43.75%
STATHEART	44.04%	39.65%	27.25%	20.71%
STUDENT	5.13%	7.14%	5.61%	6.66%
TRANSFUSION	25.68%	25.84%	24.87%	24.56%
WDBC	35.35%	29.91%	8.56%	6.43%
WINE	29.40%	59.71%	19.20%	15.86%
Z_F_S	47.81%	39.37%	10.73%	10.88%
ZO_NF_S	47.43%	43.04%	21.54%	9.76%
ZONF_S	11.99%	15.62%	4.36%	2.68%
ZOO	14.13%	10.70%	9.50%	7.27%
AVERAGE	34.48%	34.34%	26.55%	21.52%

. Pairwise significance tests on the error data yielded p = **** (i.e., p < 0.0001) for BHO versus each comparator, as shown in Figure 14. These findings indicate that BHO’s exploration and healing dynamics transfer beyond synthetic benchmarks to supervised training objectives, while a small number of datasets still favor gradient-based or classical evolutionary baselines, which motivates broader architectures and hyperparameter sweeps in future work.

5. Future Research Directions

Future work can first aim at a clearer theory of convergence under stochastic, non-stationary dynamics. A practical route is to model the algorithm’s evolution (population, incumbent best, scar/momentum/bandage) as an extended state and study conditions that ensure stability when injury intensity decays over time. Within this lens, it is important to characterize when heavy-tailed Lévy jumps accelerate basin escape without hurting late-stage stability, relative to small Gaussian steps. In parallel, an explicit complexity account will clarify per-iteration time and memory costs, and indicate when objective evaluations dominate bookkeeping overhead.

A second priority is mechanism attribution and reduced hyperparameter sensitivity. Budget-matched ablations can reveal when RAGE/hyper-RAGE, hot-dims, or Lévy steps dominate and under which landscape types they are most effective (smooth, ill-conditioned, multimodal, constrained). Building on this, self-adaptation can be introduced: Intensity and probability knobs (e.g., wp, hr) can be adjusted online from simple performance signals, while burst triggers and hot-dim boosts are governed by lightweight feedback policies, maintaining stable performance without meticulous manual tuning.

Next, scaling to high dimensions and handling realistic constraints call for targeted extensions. Subspace search, scar-weighted directions, and lightweight preconditioning (diagonal or low-rank) can improve robustness and efficiency under strong anisotropy. For constraints, noise, and non-stationarity, healing can incorporate projection/prox steps, acceptance can be made noise-robust with budgeted re-evaluations, and bursts can be triggered by simple change detectors, maintaining safety within bounded domains.

Finally, a broader evaluation program will strengthen generality. Useful steps include larger test families with budget sweeps, convergence curves that report central tendency and dispersion, and time-to-target metrics. Incorporating representative real-world cases (e.g., Lennard–Jones, economic dispatch, materials models) under identical budgets and with reproducible code/environments will clarify where BHO excels, where classical baselines such as CMA-ES or UDE-type methods are preferable, and when hybrids are the right choice. Stating limitations and contraindications alongside results will guide responsible deployment in practical settings.