# Towards a Generalised Metaheuristic Model for Continuous Optimisation Problems

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## Abstract

**:**

## 1. Introduction

- Presents the first approach to a formal mathematical model to describe a metaheuristic in terms of building blocks for tackling continuous optimisation problems. It provides standardisation for researchers to easily identify the similarities and differences between metaheuristics, preventing researchers from reinventing the wheel.
- Highlights the effectiveness of combining building blocks to generate metaheuristics to provide problem-specific solutions.
- Yields a methodology to investigate the effect of search operators.

## 2. Theoretical Foundations

#### 2.1. Optimisation

**Definition**

**1**

**Remark**

**1**

**Remark**

**2**

**Remark**

**3**

#### 2.2. Heuristics

**Definition**

**2**

**Definition**

**3**

## 3. Formulation

#### 3.1. Simple Heuristics

**Definition**

**4**

**Definition**

**5**

- Composition: $\forall \phantom{\rule{4pt}{0ex}}{h}_{1},{h}_{2}\in {\mathfrak{H}}_{i},{h}_{1}\circ {h}_{2}\in {\mathfrak{H}}_{i}$;
- Last-hit: $\forall \phantom{\rule{4pt}{0ex}}{h}_{1},{h}_{2}\in {\mathfrak{H}}_{i},\phantom{\rule{4pt}{0ex}}{h}_{1}\circ {h}_{2}={h}_{1}$;
- Associative: $\forall \phantom{\rule{4pt}{0ex}}{h}_{1},{h}_{2},{h}_{3}\in \mathfrak{H}{}_{i},\phantom{\rule{4pt}{0ex}}({h}_{1}\circ {h}_{2})\circ {h}_{3}={h}_{1}\circ ({h}_{2}\circ {h}_{3})$;
- Non-commutative: $\forall \phantom{\rule{4pt}{0ex}}{h}_{1},{h}_{2}\in \mathfrak{H}{}_{i},\phantom{\rule{4pt}{0ex}}{h}_{1}\circ {h}_{2}\ne {h}_{2}\circ {h}_{1}$.

**Remark**

**4**

**Definition**

**6**

- Composition: $\forall \phantom{\rule{4pt}{0ex}}{h}_{1},{h}_{2}\in {\mathfrak{H}}_{o},{h}_{1}\circ {h}_{2}\in {\mathfrak{H}}_{o}$;
- Identity element: $\forall \phantom{\rule{4pt}{0ex}}{h}_{1}\in {\mathfrak{H}}_{o},\phantom{\rule{4pt}{0ex}}{h}_{1}\circ {h}_{e}={h}_{e}\circ {h}_{1}={h}_{1}$;
- Inverse element: $\forall \phantom{\rule{4pt}{0ex}}{h}_{1}\in {\mathfrak{H}}_{o},\phantom{\rule{4pt}{0ex}}\exists \phantom{\rule{4pt}{0ex}}{h}_{1}^{-1}:{h}_{1}\circ {h}_{1}^{-1}={h}_{e}$;
- Associative: $\forall \phantom{\rule{4pt}{0ex}}{h}_{1},{h}_{2},{h}_{3}\in {\mathfrak{H}}_{o},\phantom{\rule{4pt}{0ex}}({h}_{1}\circ {h}_{2})\circ {h}_{3}={h}_{1}\circ ({h}_{2}\circ {h}_{3})$;
- Non-commutative: $\forall \phantom{\rule{4pt}{0ex}}{h}_{1},{h}_{2}\in {\mathfrak{H}}_{o},\phantom{\rule{4pt}{0ex}}{h}_{1}\circ {h}_{2}\ne {h}_{2}\circ {h}_{1}$.

**Remark**

**5**

**Remark**

**6**

**Definition**

**7**

**Remark**

**7**

#### 3.2. First Approach to a Generalised Metaheuristic Model

**Definition**

**8**

**Remark**

**8.**

## 4. Analysis of Selected Metaheuristics

**Definition**

**9**

**Definition**

**10**

**Definition**

**11**

**Definition**

**12**

#### 4.1. Simulated Annealing (SA)

**Definition**

**13**

#### 4.2. Particle Swarm Optimisation (PSO)

**Definition**

**14**

**Remark**

**9**

- Constrained approach: ${\alpha}_{0}={\alpha}_{1}={\alpha}_{2}=\chi $ since χ is known as the constriction factor determined by$$\chi =\frac{2\kappa H(\varphi -4)}{\varphi -2-\sqrt{\varphi (\varphi -4)}}+\sqrt{\kappa}\left(1-H(\varphi -4)\right),$$

#### 4.3. Genetic Algorithm (GA)

**Definition**

**15**

**Remark**

**10**

- Random pairing:$${z}_{i}\sim \mathcal{U}\{1,M\}$$
- Rank weighting pairing:$${z}_{i}\sim {\mathcal{U}}_{{P}_{R}}\{1,M\}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4.pt}{0ex}}\mathrm{with}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{0.166667em}{0ex}}{P}_{R}\left(m\right)=\frac{2(M+1-m)}{(M+1)M},\phantom{\rule{4pt}{0ex}}\forall \phantom{\rule{0.166667em}{0ex}}m\in \{1,\cdots ,M\}$$
- Roulette wheel pairing:$${z}_{i}\sim {\mathcal{U}}_{{P}_{C}}\{1,M\}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4.pt}{0ex}}\mathrm{with}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{0.166667em}{0ex}}{P}_{C}(m)=\frac{|f({\widehat{\overrightarrow{x}}}_{m})-f({\widehat{\overrightarrow{x}}}_{M+1})|}{\left|{\sum}_{k=1}^{M}f({\widehat{\overrightarrow{x}}}_{k})\right|},\forall \phantom{\rule{0.166667em}{0ex}}m\in \{1,\cdots ,M\}$$
- Tournament pairing:$${z}_{i}={w}_{k}\in {W}_{T}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4.pt}{0ex}}\mathrm{with}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{0.166667em}{0ex}}k\sim {\mathcal{U}}_{{P}_{T}}\{1,{M}_{T}\},\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}{P}_{T}\left(k\right)={p}_{T}{(1-{p}_{T})}^{k},$$since ${W}_{T}=\{{w}_{j}\sim \mathcal{U}\{1,M\}|(\forall \phantom{\rule{0.166667em}{0ex}}j=1,\cdots ,{M}_{T})\wedge ({w}_{1}<\cdots <{w}_{{M}_{T}})\}$ and ${p}_{T}\in (0,1]$.

**Remark**

**11**

- Single-point crossover:$$\overrightarrow{m}=H({d}_{1}-\overrightarrow{s})\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4.pt}{0ex}}\mathrm{with}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{0.166667em}{0ex}}{d}_{1}\sim \mathcal{U}\{1,D\}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{0.166667em}{0ex}}\overrightarrow{s}={(1,2,\cdots ,D)}^{\u22ba}$$
- Two-points crossover:$$\overrightarrow{m}=H({d}_{1}-\overrightarrow{s})-H({d}_{2}-\overrightarrow{s})\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4.pt}{0ex}}\mathrm{with}\phantom{\rule{4.pt}{0ex}}{d}_{1}\sim \mathcal{U}\{1,D\},\phantom{\rule{0.166667em}{0ex}}{d}_{2}\sim \mathcal{U}\{{d}_{1}+1,D\},\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}\overrightarrow{s}={(1,2,\cdots ,D)}^{\u22ba}$$
- Uniform crossover:$$\overrightarrow{m}=H(\overrightarrow{r}-0.5)\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4.pt}{0ex}}\mathrm{with}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{0.166667em}{0ex}}\overrightarrow{r}\ni {r}_{i}\sim \mathcal{U}(0,1)$$
- Blend crossover:$$\overrightarrow{m}=\overrightarrow{r}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4.pt}{0ex}}\mathrm{with}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{0.166667em}{0ex}}\overrightarrow{r}\ni {r}_{i}\sim \mathcal{U}(0,1)$$
- Linear crossover:$$\overrightarrow{m}={\beta}_{1}\overrightarrow{1}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{0.166667em}{0ex}}1-\overrightarrow{m}={\beta}_{2}\overrightarrow{1}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4.pt}{0ex}}\mathrm{with}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{0.166667em}{0ex}}{\beta}_{1},{\beta}_{2}\in \mathbb{R}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{0.166667em}{0ex}}\overrightarrow{1}\in {\mathbb{Z}}_{1}^{D}$$

**Definition**

**16**

#### 4.4. Differential Evolution (DE)

**Definition**

**17**

**Remark**

**12**

- rand:$$\overrightarrow{m}={\overrightarrow{x}}_{{z}_{0}}\left(t\right)$$
- best:$$\overrightarrow{m}={\overrightarrow{x}}_{*}\left(t\right)$$
- current-to-best:$$\overrightarrow{m}={\overrightarrow{x}}_{n}\left(t\right)+{\alpha}_{0}({\overrightarrow{x}}_{*}\left(t\right)-{\overrightarrow{x}}_{{z}_{0}}\left(t\right))$$
- rand-to-best-and-current:$$\overrightarrow{m}={\overrightarrow{x}}_{{z}_{0}}\left(t\right)+{\alpha}_{0}({\overrightarrow{x}}_{*}\left(t\right)-{\overrightarrow{x}}_{n}\left(t\right))$$

**Definition**

**18**

**Remark**

**13**

- Binomial crossover:$${p}_{\mathrm{CR}}=\mathrm{CR}\left(1-\frac{1}{D}\right)+\frac{1}{D}$$
- Exponential crossover:$${p}_{\mathrm{CR}}=\frac{1-{\mathrm{CR}}^{D}}{D(1-\mathrm{CR})}$$

#### 4.5. Summary

## 5. Numerical Examples

## 6. Summary

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

DC | Differential Crossover |

DE | Differential Evolution |

DM | Differential Mutation |

GA | Genetic Algorithm |

GC | Genetic Crossover |

GM | Genetic Mutation |

MH | Metaheuristic |

PS | Particle Swarm |

PSO | Particle Swarm Optimisation |

RS | Random Search |

SA | Simulated Annealing |

SH | Simple Heuristic |

SO | Search Operator |

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**Figure 1.**Block-diagram of a metaheuristic composed by an initialiser ${h}_{i}$, $\varpi $ perturbators ${h}_{j,p}$ and selectors ${h}_{j,s}$, $\forall \phantom{\rule{4pt}{0ex}}j\in \{1,\cdots ,\varpi \}$, and a finaliser ${h}_{f}$.

**Figure 2.**Example of (

**a**) the metaheuristic representation based on signal-flow diagrams, where ${h}_{o}$ can be obtained from (

**b**) a cascade or (

**c**) a parallel composition.

**Figure 3.**Representation of the selected metaheuristics using the proposed model and their search operators (perturbators and selectors).

**Figure 4.**Violin-plots of the fitness orders, ${log}_{10}\left(f\left({\overrightarrow{x}}_{*}\right)\right)$, obtained for each metaheuristic built when solving a given problem. Mean and median are indicated by the red and green vertical strokes, respectively.

**Figure 5.**Rank of the metaheuristics based on the median and interquartile range of the fitness values obtained when solving Sphere, Stochastic, and Schaffer N3, both for 2, 10, and 50 dimensions.

**Figure 6.**Fitness evolution for some selected metaheuristics solving the problems Sphere 2D, Stochastic 10D, and Schaffer N3 50D. Green, blue, and red strokes are used for the best, mid, and lowest ranked solvers, respectively.

Simple Heuristic | Symbol | Element | Definition |
---|---|---|---|

Initialiser | ${h}_{i}$ | 5 | |

Search Operator | ${h}_{o}$ | 6 | |

Finaliser | ${h}_{f}$ | 7 |

**Table 2.**Search operators extracted from four well-known metaheuristics in the literature. Values or ranges for variation and tuning parameters, as well as selectors, are chosen from those commonly employed in the literature.

Perturbation Heuristic | Def. | Variation Parameters | Tuning Parameters | Selection Heuristic |
---|---|---|---|---|

Random Search, ${h}_{\mathrm{RS}}$ | 13 | $\overrightarrow{r}\ni {r}_{i}\sim \mathcal{U}(-1,1)$ | $\alpha \in (0,1]$ | Metropolis, ${h}_{\mathrm{M}}$ (Definition 11) |

Swarm Dynamic, ${h}_{\mathrm{PS}}$ | 14 | Velocity approach, ${\overrightarrow{r}}_{i}\ni {r}_{i,j}\sim \mathcal{U}(0,1)\phantom{\rule{0.166667em}{0ex}}\forall \phantom{\rule{0.166667em}{0ex}}i\in \{1,2\}$ | ${\alpha}_{0}\in (0,1)$, ${\varphi}_{1},{\varphi}_{2}\in (0,4]$, $\kappa \in [0,1]$ | Direct, ${h}_{\mathrm{D}}$ (Definition 9) |

Genetic Crossover, ${h}_{\mathrm{GC}}$ | 15 | Pairing Scheme ^{a}, Crossover Mechanism ^{b} | ${m}_{p}\in (0,1]$ | Direct, ${h}_{\mathrm{D}}$ (Definition 9) |

Genetic Mutation, ${h}_{\mathrm{GM}}$ | 16 | $\overrightarrow{r}\ni {r}_{i}\sim \mathcal{U}(-1,1)$ | $\alpha \in (0,1]$, ${p}_{e}\in [0,1]$, ${p}_{m}\in (0,1]$ | Greedy, ${h}_{\mathrm{G}}$ (Definition 10) |

Differential Mutation, ${h}_{\mathrm{DM}}$ | 17 | Mutation Scheme | ${\alpha}_{m}\in (0,3)\phantom{\rule{4pt}{0ex}}\forall \phantom{\rule{0.166667em}{0ex}}m\in \{0,M\}$, $M\in {\mathbb{Z}}_{+}$ | Direct, ${h}_{\mathrm{D}}$ (Definition 9) |

Differential Crossover, ${h}_{\mathrm{DC}}$ | 18 | Crossover Type ^{c} | $CR\in (0,1]$ | Greedy, ${h}_{\mathrm{G}}$ (Definition 10) |

^{a}Pairing schemes: random, rank weighting, roulette wheel, and tournament pairing. Tournament pairing requires two additional parameters such as ${M}_{T}\in \{2,3\}$ and ${p}_{T}\in (0,1]$.

^{b}Crossover mechanisms: single-point, two-points, uniform, blend, and linear crossover. Linear crossover requires two additional parameters, such as ${\beta}_{1},{\beta}_{2}\in \mathbb{R}$.

^{c}Crossover types: binomial and exponential.

Perturbation Heuristic | Variation Parameters | Tuning Parameters |
---|---|---|

Random Search, ${h}_{\mathrm{RS}}$ | $\overrightarrow{r}\ni {r}_{i}\sim \mathcal{U}(-1,1)$ | $\alpha =1.0$ |

Swarm Dynamic, ${h}_{\mathrm{PS}}$ | Inertial approach, ${\overrightarrow{r}}_{i}\ni {r}_{i,j}\sim \mathcal{U}(0,1)\phantom{\rule{0.166667em}{0ex}}\forall \phantom{\rule{0.166667em}{0ex}}i\in \{1,2\}$ | ${\alpha}_{0}=1.0$, ${\varphi}_{1}=2.54,{\varphi}_{2}=2.56$ |

Genetic Crossover, ${h}_{\mathrm{GC}}$ | Tournament Pairing with ${M}_{T}=2$ and ${p}_{T}=1.0$, Single-point Crossover | ${m}_{p}=0.4$ |

Genetic Mutation, ${h}_{\mathrm{GM}}$ | $\overrightarrow{r}\ni {r}_{i}\sim \mathcal{U}(-1,1)$ | $\alpha =1.0$, ${p}_{e}=0.1$, ${p}_{m}=0.25$ |

Differential Mutation, ${h}_{\mathrm{DM}}$ | Scheme DE/current-to-best/ | ${\alpha}_{m}=1.0,\phantom{\rule{4pt}{0ex}}\forall \phantom{\rule{0.166667em}{0ex}}m\in \{0,M\}$, $M=1$ |

Differential Crossover, ${h}_{\mathrm{DC}}$ | Binomial Crossover | $CR=0.2$ |

Conf. | Search Operator, ${\mathit{h}}_{\mathit{o}}$ | Name | |
---|---|---|---|

${\mathbf{h}}_{\mathbf{o},\mathbf{1}}$ | ${\mathbf{h}}_{\mathbf{o},\mathbf{2}}$ | ||

1 | ${h}_{\mathrm{RS}}\circ {h}_{\mathrm{M}}$ | – | SA (or RSm) |

2 | ${h}_{\mathrm{PS}}\circ {h}_{\mathrm{D}}$ | – | PSO (or PSd) |

3 | ${h}_{\mathrm{GC}}\circ {h}_{\mathrm{D}}$ | – | GCd |

4 | ${h}_{\mathrm{GM}}\circ {h}_{\mathrm{G}}$ | – | GMg |

5 | ${h}_{\mathrm{DM}}\circ {h}_{\mathrm{D}}$ | – | DMd |

6 | ${h}_{\mathrm{DC}}\circ {h}_{\mathrm{G}}$ | – | DCg |

7 | ${h}_{\mathrm{RS}}\circ {h}_{\mathrm{G}}$ | – | RSg |

8 | ${h}_{\mathrm{PS}}\circ {h}_{\mathrm{M}}$ | – | PSm |

9 | ${h}_{\mathrm{GC}}\circ {h}_{\mathrm{M}}$ | – | GCm |

10 | ${h}_{\mathrm{DM}}\circ {h}_{\mathrm{M}}$ | – | DMm |

11 | ${h}_{\mathrm{GC}}\circ {h}_{\mathrm{D}}$ | ${h}_{\mathrm{GM}}\circ {h}_{\mathrm{G}}$ | GA (or GCd-GMg) |

12 | ${h}_{\mathrm{DM}}\circ {h}_{\mathrm{D}}$ | ${h}_{\mathrm{DC}}\circ {h}_{\mathrm{G}}$ | DE (or DMd-DCg) |

13 | ${h}_{\mathrm{RS}}\circ {h}_{\mathrm{M}}$ | ${h}_{\mathrm{PS}}\circ {h}_{\mathrm{D}}$ | SA-PSO |

14 | ${h}_{\mathrm{PS}}\circ {h}_{\mathrm{D}}$ | ${h}_{\mathrm{RS}}\circ {h}_{\mathrm{M}}$ | PSO-SA |

15 | ${h}_{\mathrm{GC}}\circ {h}_{\mathrm{D}}$ | ${h}_{\mathrm{DC}}\circ {h}_{\mathrm{G}}$ | GCd-DCg |

16 | ${h}_{\mathrm{DM}}\circ {h}_{\mathrm{D}}$ | ${h}_{\mathrm{GM}}\circ {h}_{\mathrm{G}}$ | DMd-GMg |

17 | ${h}_{\mathrm{PS}}\circ {h}_{\mathrm{D}}$ | ${h}_{\mathrm{DC}}\circ {h}_{\mathrm{G}}$ | PSO-DCg |

18 | ${h}_{\mathrm{RS}}\circ {h}_{\mathrm{M}}$ | ${h}_{\mathrm{DC}}\circ {h}_{\mathrm{G}}$ | SA-DCg |

19 | ${h}_{\mathrm{GC}}\circ {h}_{\mathrm{D}}$ | ${h}_{\mathrm{RS}}\circ {h}_{\mathrm{M}}$ | GCd-SA |

20 | ${h}_{\mathrm{PS}}\circ {h}_{\mathrm{D}}$ | ${h}_{\mathrm{DC}}\circ {h}_{\mathrm{M}}$ | PSO-DCm |

Function | Expression, $\mathit{f}\left(\overrightarrow{\mathit{x}}\right):{\mathbb{R}}^{\mathit{D}}\mapsto \mathbb{R}$ | Domain, $\mathfrak{X}\subseteq {\mathbb{R}}^{\mathit{D}}$ | Representation in 2D |
---|---|---|---|

Sphere [40] | $f\left(\overrightarrow{x}\right)=\sum _{i=1}^{D}{x}_{i}^{2}$ | ${[-100,100]}^{D}$ | |

Stochastic [41] | $f\left(\overrightarrow{x}\right)=\sum _{i=1}^{D}{r}_{i}\left|{x}_{i}-\frac{1}{i}\right|,\phantom{\rule{4pt}{0ex}}{r}_{i}\sim \mathcal{U}(0,1)$ | ${[-5,5]}^{D}$ | |

Schaffer N3 [42] | $f\left(\overrightarrow{x}\right)=\frac{D-1}{2}+\sum _{i=1}^{D-1}\frac{{sin}^{2}\left(cos|{x}_{i}+{x}_{i+1}|\right)-0.5}{{(1+0.001({x}_{i}+{x}_{i-1}))}^{2}}$ | ${[-100,100]}^{D}$ |

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**MDPI and ACS Style**

Cruz-Duarte, J.M.; Ortiz-Bayliss, J.C.; Amaya, I.; Shi, Y.; Terashima-Marín, H.; Pillay, N.
Towards a Generalised Metaheuristic Model for Continuous Optimisation Problems. *Mathematics* **2020**, *8*, 2046.
https://doi.org/10.3390/math8112046

**AMA Style**

Cruz-Duarte JM, Ortiz-Bayliss JC, Amaya I, Shi Y, Terashima-Marín H, Pillay N.
Towards a Generalised Metaheuristic Model for Continuous Optimisation Problems. *Mathematics*. 2020; 8(11):2046.
https://doi.org/10.3390/math8112046

**Chicago/Turabian Style**

Cruz-Duarte, Jorge M., José C. Ortiz-Bayliss, Iván Amaya, Yong Shi, Hugo Terashima-Marín, and Nelishia Pillay.
2020. "Towards a Generalised Metaheuristic Model for Continuous Optimisation Problems" *Mathematics* 8, no. 11: 2046.
https://doi.org/10.3390/math8112046