# From an Optimal Point to an Optimal Region: A Novel Methodology for Optimization of Multimodal Constrained Problems and a Novel Constrained Sliding Particle Swarm Optimization Strategy

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## Abstract

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## 1. Introduction

- A statistical test to evaluate the likelihood confidence regions from constrained meta-heuristic optimization is proposed;
- A novel constraints-based sliding particle swarm optimizer is presented to address multi-local minima problems;
- The proposed methodology is evaluated using several benchmark tests and using the optimization problem of a chemical process as a practical case study.

## 2. Materials and Methods

#### 2.1. Likelihood Confidence Region of Constrained Optimization Problems

#### 2.2. A Sliding Particle Swarm Optimization for Solving Constrained Optimization Problems

Algorithm 1. A pseudo-code of the proposed CSPSO |

BeginSet PSO parameters Acceleration coefficient 1:= ${c}_{1}$ Acceleration coefficient 2:=${c}_{2}$ Define the value of the criteria χ for the maximum number of iteration within the same minima $\mathrm{Define}\mathrm{range}\alpha $ of a local minima l = 0 while (termination condition = false, expansion of search domain)$\mathrm{Initialize}\mathrm{the}\mathrm{population}\mathrm{within}\mathrm{search}\mathrm{region},\left({R}_{l,min},{R}_{l,max}\right)$ $\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{v}_{p}^{}=0$ $\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{x}_{p}^{}={R}_{l,max}+rnd\left({R}_{l,max}-{R}_{l,min}\right)$ $\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{v}_{max,p}=\left({X}_{p,max}-{X}_{p,min}\right)/2$ $\hspace{1em}\hspace{1em}\hspace{1em}\mathbf{for}j=1$ to number of iterations $\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathbf{for}\mathrm{p}=1$ to number of particles $\mathbf{if}\Delta =0$ $\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{evaluate}\mathrm{objective}\mathrm{function}\mathrm{with}\mathrm{penalties}:={f}_{j}$$(C,{x}_{p}^{}$) $\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathbf{for}i=1$ to number of dimensions update particle position and velocity $\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{v}_{p}^{i}={c}_{1}ran{d}_{1}\left({x}_{p}^{per}-{x}_{p}^{i}\right)+{c}_{2}ran{d}_{2}\left({x}_{p}^{glo}-{x}_{p}^{i}\right)$ $\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathbf{if}{v}_{p}^{i}{v}_{max,p}$ $\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{v}_{p}^{k+1}={v}_{max}sign\left({v}_{p}^{k+1}\right)$ $\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathbf{\text{endif}}$ $\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{x}_{p}^{i}={x}_{p}^{i}+{v}_{p}^{i}$ $\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathbf{if}{x}_{p}^{i}{R}_{l,max}$ $\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{x}_{p}^{i}=\frac{({R}_{i,min}-{R}_{i,\mathrm{max}})}{2}$ $\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathbf{\text{endif}}$ $\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathbf{if}{x}_{p}^{i}{X}_{p,min}$ $\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{x}_{p}^{i}=\frac{({R}_{i,min}-{R}_{i,\mathrm{max}})}{2}$ $\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathbf{\text{end if}}$ $\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathbf{\text{end for}}$ $\mathbf{else}$ $\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathbf{while}{f}_{j}\left(C,{x}_{p}^{}\right){\mathsf{\delta}}_{}$ $\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{evaluate}\mathrm{objective}\mathrm{function}:={V}_{k}$$({x}_{p}^{}$) $\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathbf{for}i=1$ to number of dimensions $\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{update}\mathrm{particle}\mathrm{position}\mathrm{and}\mathrm{velocity}$ $\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{v}_{p}^{i}={c}_{1}ran{d}_{1}\left({x}_{p}^{per}-{x}_{p}^{i}\right)+{c}_{2}ran{d}_{2}\left({x}_{p}^{glo}-{x}_{p}^{i}\right)$ $\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathbf{if}{v}_{p}^{i}{v}_{max,p}$ $\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{v}_{p}^{k+1}={v}_{max}sign\left({v}_{p}^{k+1}\right)$ $\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathbf{\text{end if}}$ $\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{x}_{p}^{i}={x}_{p}^{i}+{v}_{p}^{i}$ $\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathbf{if}{x}_{p}^{i}{R}_{l,max}$ $\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{x}_{p}^{i}=\frac{({R}_{i,min}-{R}_{i,\mathrm{max}})}{2}$ $\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathbf{\text{end if}}$ $\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathbf{if}{x}_{p}^{i}{X}_{p,min}$ $\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{x}_{p}^{i}=\frac{({R}_{i,min}-{R}_{i,\mathrm{max}})}{2}$ $\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathbf{\text{end if}}$ $\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathbf{\text{end for}}$ $\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathbf{end}$ $\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathbf{\text{end if}}$ $\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathbf{\text{end for}}$ $\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathbf{\text{end for}}$ $:{x}_{p,l}^{glo}={x}_{p}^{glo}$ $\mathrm{if}\mathrm{cont}\mathsf{\chi}\mathrm{and}\left(1-\alpha \right){x}_{p,l-1}^{glo}{x}_{p,l}^{glo}\alpha {x}_{p,l-1}^{glo}$ ${R}_{l+1,min}={R}_{l,min}\ast expansionfactor$ $l=l+1$ end |

## 3. Results

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The resulting space formed by the overall population originated by the algorithm. The relationship between the value of each particle with a theoretical global minimum is expressed in terms of Pythagorean theorem. Based on this theorem it is presented a schematical representation of the concepts used in the deductions of the Fisher–Snedecor test.

**Figure 2.**Schematic representation of the possible search spaces (${R}_{1},{R}_{2}$, and ${R}_{3}$) of a given function with ${L}_{1},{L}_{2},$ and ${L}_{4}$ as local minima and ${L}_{3}$ and ${L}_{5}$ as constrained minima.

**Figure 4.**Ackley Function 3D topography drawn by the proposed CSPSO and its corresponding 2D map of the local and global minima, with the confidence regions drawn by the deduced likelihood test. It is also presented a surface map of the objective function in order to show the location and values of the minima and global minimum.

**Figure 5.**Rastrigin Function 3D topography drawn by the proposed CSPSO and its corresponding 2D map of the local and global minima, with the confidence regions drawn by the deduced likelihood test. A surface map of the objective function is also presented in order to show the location and values of all minima.

**Figure 6.**Cross-in-Tray Function 3D topography drawn by the proposed CSPSO and its corresponding 2D map of the local and global minima, with the confidence regions drawn by the deduced likelihood test. A surface map of the objective function is also presented in order to show the location and values of the minima.

**Figure 7.**Egg Crate Function with input and output constraints 3D topography drawn by the proposed CSPSO and its corresponding 2D map of the local and global minima, with the confidence regions drawn by the deduced likelihood test. A surface map of the objective function is also presented in order to show the location and values of the minima and global minimum.

**Figure 8.**Keane Function constrained with input and output constraints 3D topography drawn by the proposed CSPSO and its corresponding 2D map of the local and global minima, with the confidence regions drawn by the deduced likelihood test. A surface map of the objective function is also presented in order to show the location and values of the minima and global minimum.

**Figure 9.**Himmelblau Function 3D topography drawn by the proposed CSPSO and its corresponding 2D map of the local and global minima, with the confidence regions drawn by the deduced likelihood test. A surface map of the objective function is also presented in order to show the location and values of the minima and global minimum.

**Figure 10.**Process optimization results based on the proposed methodology. At the top of the figure is presented the objective function landscape, then it is presented the operating confidence region defined by the proposed test and the contour area where the local minima are located.

Mesh number Number of particles for each mesh Number of iterations for each mesh Number of parameters ${C}_{1i}$ ${C}_{1f}$ ${C}_{2i}$ ${C}_{2f}$ Confidence level | 20 100 100 2 0.5 2.5 2.5 0.5 0.99 |

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**MDPI and ACS Style**

Rebello, C.M.; Martins, M.A.F.; Loureiro, J.M.; Rodrigues, A.E.; Ribeiro, A.M.; Nogueira, I.B.R. From an Optimal Point to an Optimal Region: A Novel Methodology for Optimization of Multimodal Constrained Problems and a Novel Constrained Sliding Particle Swarm Optimization Strategy. *Mathematics* **2021**, *9*, 1808.
https://doi.org/10.3390/math9151808

**AMA Style**

Rebello CM, Martins MAF, Loureiro JM, Rodrigues AE, Ribeiro AM, Nogueira IBR. From an Optimal Point to an Optimal Region: A Novel Methodology for Optimization of Multimodal Constrained Problems and a Novel Constrained Sliding Particle Swarm Optimization Strategy. *Mathematics*. 2021; 9(15):1808.
https://doi.org/10.3390/math9151808

**Chicago/Turabian Style**

Rebello, Carine M., Márcio A. F. Martins, José M. Loureiro, Alírio E. Rodrigues, Ana M. Ribeiro, and Idelfonso B. R. Nogueira. 2021. "From an Optimal Point to an Optimal Region: A Novel Methodology for Optimization of Multimodal Constrained Problems and a Novel Constrained Sliding Particle Swarm Optimization Strategy" *Mathematics* 9, no. 15: 1808.
https://doi.org/10.3390/math9151808