Fine-Tuned Constrained Nelder–Mead SOMA

Abstract

A number of population-based constrained optimization techniques are offered in the literature. Usually, a least amount of 50 population size is required to solve constrained optimization problems proficiently that results in large computation cost also. In this paper, a hybrid optimization technique, constrained Nelder–Mead self-organizing migrating algorithm (C-NMSOMA), has been projected that works with 20 population size only. It works with aim not only to handle constraints but also to condense the computation cost also. To confirm the claim, an experiment has been conducted for a set of ten constraint optimization problems by varying the population size from 20 to 100. C-NMSOMA works best with population size 20. To show its effectiveness over other algorithms, a convincing comparison has been made between the best results available by these algorithms and results obtained by C-NMSOMA using population size 20 only. Experimental results demonstrate that the presented algorithm C-NMSOMA is a robust optimization technique that yields feasible and viable solutions in lesser number of function evaluations using lesser population size. It combines the features of self-organizing migrating algorithm (SOMA) and Nelder–Mead (NM) simplex search. NM simplex search has been used as a crossover operator to produce new individuals in the solution space. A constraint handling technique based on preserving the feasibility of solutions with initialized feasible population has been adopted.

Appendix

1.1 Constrained Problems

The first six functions are well-known benchmark taken from Runarsson and Yao [43].

Problem 01

$$ \begin{array}{*{20}l} {\text{Minimize}} \hfill & {\text{f} \left( {\text{x}} \right) = 5\sum\limits_{{{\text{i}} = 1}}^{4} {{\text{x}}_{\text{i}} - 5} \sum\limits_{{{\text{i}} = 1}}^{4} {{\text{x}}_{{_{\text{i}} }}^{2} - } \sum\limits_{{{\text{i}} = 5}}^{13} {{\text{x}}_{\text{i}} } } \hfill \\ {{\text{subject}}\,{\text{to:}}} \hfill & {\text{g}_{1} \left( {\text{x}} \right) = 2{\text{x}}_{1} + 2{\text{x}}_{2} + {\text{x}}_{10} + {\text{x}}_{11} - 10 \le 0} \hfill \\ {} \hfill & {\text{g}_{2} \left( {\text{x}} \right) = 2{\text{x}}_{1} + 2{\text{x}}_{3} + {\text{x}}_{10} + {\text{x}}_{12} - 10 \le 0} \hfill \\ {} \hfill & {\text{g}_{3} \left( {\text{x}} \right) = 2{\text{x}}_{2} + 2{\text{x}}_{3} + {\text{x}}_{11} + {\text{x}}_{12} - 10 \le 0} \hfill \\ {} \hfill & {\text{g}_{4} \left( {\text{x}} \right) = - 8{\text{x}}_{1} + {\text{x}}_{10} \le 0} \hfill \\ {} \hfill & {\text{g}_{5} \left( {\text{x}} \right) = - 8{\text{x}}_{2} + {\text{x}}_{11} \le 0} \hfill \\ {} \hfill & {\text{g}_{6} \left( {\text{x}} \right) = - 8{\text{x}}_{3} + {\text{x}}_{12} \le 0} \hfill \\ {} \hfill & {\text{g}_{7} \left( {\text{x}} \right) = - 2{\text{x}}_{4} - {\text{x}}_{5} + {\text{x}}_{10} \le 0} \hfill \\ {} \hfill & {\text{g}_{8} \left( {\text{x}} \right) = - 2{\text{x}}_{6} - {\text{x}}_{7} + {\text{x}}_{11} \le 0} \hfill \\ {} \hfill & {\text{g}_{9} \left( {\text{x}} \right) = - 2{\text{x}}_{8} - {\text{x}}_{9} + {\text{x}}_{12} \le 0} \hfill \\ \end{array} $$

$0 \le {\text{x}}_{\text{i}} \le 1$, i = 1, .., 9; $0 \le {\text{x}}_{\text{i}} \le 100$, i = 10, 11, 12; $0 \le {\text{x}}_{13} \le 1$.

The global minimum optimal solution is:

$$ {\bar{\text{x}}} = \left( {1,1,1,1,1,1,1,1,1,3,3,3,1} \right),f\left( {{\bar{\text{x}}}} \right) = - 15 $$

Problem 02

$$ \begin{array}{*{20}l} {\text{Maximize }} \hfill & {{\text{f}}\left( {\text{x}} \right) = \left| {\frac{{\sum\limits_{{{\text{i}} = 1}}^{\text{n}} {{ \cos }^{4} \left( {{\text{x}}_{\text{i}} } \right) - 2\prod\limits_{{{\text{i}} = 1}}^{\text{n}} {{ \cos }^{2} \left( {{\text{x}}_{\text{i}} } \right)} } }}{{\sqrt {\sum\limits_{{{\text{i}} = 1}}^{\text{n}} {{\text{ix}}_{\text{i}}^{2} } } }}} \right|} \hfill \\ {{\text{Subject}}\,{\text{to:}}} \hfill & {{\text{g}}_{1} \left( {\text{x}} \right) = 0.75 - \prod\limits_{{{\text{i}} = 1}}^{\text{n}} {{\text{x}}_{\text{i}} } \le 0} \hfill \\ {} \hfill & {{\text{g}}_{2} \left( {\text{x}} \right) = \sum\limits_{{{\text{i}} = 1}}^{\text{n}} {{\text{x}}_{\text{i}} } - 7.5\,{\text{n}} \le 0} \hfill \\ \end{array} $$

where n = 20 and $0 \le {\text{x}}_{\text{i}} \le 10$, i = 1, …, n.

The global maximum optimal solution is:

$$ f\left( {\bar{x}} \right) = 0.803619. $$

Problem 03

$$ \begin{array}{*{20}l} {\text{Maximize}} \hfill & {\text{f} \left( {\text{x}} \right) = \left( {\sqrt {\text{n}} } \right)^{\text{n}} \prod\limits_{{{\text{i}} = 1}}^{\text{n}} {{\text{x}}_{\text{i}} } } \hfill \\ {{\text{Subject}}\,{\text{to:}}} \hfill & {\text{h} \left( {\text{x}} \right) = \sum\limits_{{{\text{i}} = 1}}^{\text{n}} {{\text{x}}_{\text{i}}^{2} } - 1 = 0} \hfill \\ \end{array} $$

where n= 10 and $0 \le {\text{x}}_{\text{i}} \le 1$, i = 1, …, n.

The global maximum optimal solution is:

$$ \bar{x} = \left( {\frac{1}{\sqrt n }, \ldots ,\frac{1}{\sqrt n }} \right),f\left( {\bar{x}} \right) = 1. $$

Problem 04

$$ \begin{array}{*{20}l} {Minimize} \hfill & {\text{f} \left( {\text{x}} \right) = 5.3578547\,{\text{x}}_{3}^{2} + 0.8356891\,{\text{x}}_{1} {\text{x}}_{5} + 37.293239\,{\text{x}}_{1} - 40792.141} \hfill \\ {{\text{Subject}}\,{\text{to:}}} \hfill & {\text{g}_{1} \left( {\text{x}} \right) = 85.334407 + 0.0056858\,{\text{x}}_{2} {\text{x}}_{5} + 0.0006262\,{\text{x}}_{1} {\text{x}}_{4} - 0.0022053\,{\text{x}}_{3} {\text{x}}_{5} - 92 \le 0} \hfill \\ {} \hfill & {\text{g}_{2} \left( {\text{x}} \right) = - 85.334407 - 0.0056858\,{\text{x}}_{2} {\text{x}}_{5} - 0.0006262\,{\text{x}}_{1} {\text{x}}_{4} + 0.0022053\,{\text{x}}_{3} {\text{x}}_{5} \le 0} \hfill \\ {} \hfill & {\text{g}_{3} \left( {\text{x}} \right) = 80.51249 + 0.0071317\,{\text{x}}_{2} {\text{x}}_{5} + 0.0029955\,{\text{x}}_{1} {\text{x}}_{2} + 0.0021813\,{\text{x}}_{3}^{2} - 110 \le 0} \hfill \\ {} \hfill & {\text{g}_{4} \left( {\text{x}} \right) = - 80.51249 - 0.0071317\,{\text{x}}_{2} {\text{x}}_{5} - 0.0029955\,{\text{x}}_{1} {\text{x}}_{2} - 0.0021813\,{\text{x}}_{3}^{2} + 90 \le 0} \hfill \\ {} \hfill & {\text{g}_{5} \left( {\text{x}} \right) = 9.300961 + 0.0047026\,{\text{x}}_{3} {\text{x}}_{5} + 0.0012547\,{\text{x}}_{1} {\text{x}}_{3} + 0.0019085\,{\text{x}}_{3} {\text{x}}_{4} - 25 \le 0} \hfill \\ {} \hfill & {\text{g}_{6} \left( {\text{x}} \right) = - 9.300961 - 0.0047026\,{\text{x}}_{3} {\text{x}}_{5} - 0.0012547\,{\text{x}}_{1} {\text{x}}_{3} - 0.0019085\,{\text{x}}_{3} {\text{x}}_{4} + 20 \le 0} \hfill \\ {} \hfill & {78 \le {\text{x}}_{1} \le 102,33 \le {\text{x}}_{2} \le 45,27 \le {\text{x}}_{\text{i}} \le 45,\,{\text{i}} = 3,4,5.} \hfill \\ \end{array} $$

The global minimum optimal solution is:

$$ {\bar{\text{x}}} = \left( {78.0,33.0,29.995256025682,45.0,36.775812905788} \right),f\left( {{\bar{\text{x}}}} \right) = - 30665.539. $$

Problem 05

$$ \begin{array}{*{20}l} {\text{Maximize}} \hfill & {\text{f} \left( {\text{x}} \right) = \frac{{{ \sin }^{3} \left( {2\pi \,{\text{x}}_{1} } \right){ \sin }\left( {2\pi \,{\text{x}}_{2} } \right)}}{{{\text{x}}_{1}^{3} \left( {{\text{x}}_{1} + {\text{x}}_{2} } \right)}}} \hfill \\ {{\text{Subject}}\,{\text{to:}}} \hfill & {\text{g}_{1} \left( {\text{x}} \right) = {\text{x}}_{1}^{2} - {\text{x}}_{2} + 1 \le 0} \hfill \\ {} \hfill & {\text{g}_{2} \left( {\text{x}} \right) = 1 - {\text{x}}_{1} + \left( {{\text{x}}_{2} - 4} \right)^{2} \le 0} \hfill \\ {} \hfill & {0 \le {\text{x}}_{\text{i}} \le 10,\,\,\,\,\,{\text{i}} = 1,2.} \hfill \\ \end{array} $$

The global maximum optimal solution is:

$$ \bar{x} = \left( {1.2279713,4.2453733} \right),f\left( {\bar{x}} \right) = 0.095825. $$

Problem 06

$$ \begin{array}{*{20}l} {\text{Minimize}} \hfill & {\text{f} \left( {\text{x}} \right) = {\text{x}}_{1}^{2} + \left( {{\text{x}}_{2} - 1} \right)^{2} } \hfill \\ {{\text{Subject}}\,{\text{to:}}} \hfill & {\text{h} \left( {\text{x}} \right) = {\text{x}}_{2} - {\text{x}}_{1}^{2} = 0} \hfill \\ {} \hfill & { - 1 \le {\text{x}}_{\text{i}} \le 1,\,\,\,\,\,{\text{i}} = 1,2.} \hfill \\ \end{array} $$

The global minimum optimal solution is:

$$ \bar{x} = \left( { \pm \frac{1}{\sqrt 2 },\frac{1}{2}} \right),f\left( {\bar{x}} \right) = 0.75. $$

Problem 07 This problem has been taken from handbook of test problems in local and global optimizations [44].

$$ \begin{array}{*{20}l} {\text{Minimize}} \hfill & {\text{f} \left( {\text{x}} \right) = \left( { - {\text{x}}_{1} - {\text{x}}_{5} + 0.4\,{\text{x}}_{1}^{0.67} {\text{x}}_{3}^{ - 0.67} + 0.4\,{\text{x}}_{5}^{0.67} {\text{x}}_{7}^{ - 0.67} } \right)} \hfill \\ {{\text{Subject}}\,{\text{to:}}} \hfill & {\text{g}_{1} \left( {\text{x}} \right) = 0.05882\,{\text{x}}_{3} \,{\text{x}}_{4} + 0.1\,{\text{x}}_{1} \le 1} \hfill \\ {} \hfill & {\text{g}_{2} \left( {\text{x}} \right) = 0.05882\,{\text{x}}_{7} \,{\text{x}}_{8} + 0.1\,{\text{x}}_{1} + 0.1\,{\text{x}}_{5} \le 1} \hfill \\ {} \hfill & {\text{g}_{3} \left( {\text{x}} \right) = 4\,{\text{x}}_{2\,} {\text{x}}_{4}^{ - 1} + 2\,{\text{x}}_{2}^{ - 0.71} {\text{x}}_{4}^{ - 1} + 0.05882\,{\text{x}}_{2}^{ - 1.3} {\text{x}}_{3} \le 1} \hfill \\ {} \hfill & {\text{g}_{4} \left( {\text{x}} \right) = 4\,{\text{x}}_{6\,} {\text{x}}_{8}^{ - 1} + 2\,{\text{x}}_{6}^{ - 0.71} {\text{x}}_{8}^{ - 1} + 0.05882\,{\text{x}}_{6}^{ - 1.3} {\text{x}}_{7} \le 1} \hfill \\ {} \hfill & {0.01 \le {\text{x}}_{\text{i}} \le 10,\,{\text{i}} = 1,2, \ldots ,8.} \hfill \\ \end{array} $$

The optimal solution of the problem is:

$$ \begin{aligned} \bar{x} & = \left( {6.4225,\,0.6686,\,1.0239,\,5.9399,\,2.2673,\,0.5960,\,0.4029,5.5288} \right) \\ f\left( {\bar{x}} \right) & = - 6.0482. \\ \end{aligned} $$

Problem 08 This problem was proposed by Mezura and Coello [45].

$$ \begin{array}{*{20}l} {\text{Minimize}} \hfill & {\text{f} \left( {\text{x}} \right) = - {\text{x}}_{ 1} - {\text{x}}_{2} } \hfill \\ {{\text{Subject}}\,{\text{to:}}} \hfill & {\text{g}_{1} \left( {\text{x}} \right) = - 2\,{\text{x}}_{1}^{4} + 8\,{\text{x}}_{1}^{3} - 8\,{\text{x}}_{1}^{2} + {\text{x}}_{2} - 2 \le 0} \hfill \\ {} \hfill & {\text{g}_{2} \left( {\text{x}} \right) = - 4\,{\text{x}}_{1}^{4} + 32\,{\text{x}}_{1}^{3} - 88\,{\text{x}}_{1}^{2} + 96\,{\text{x}}_{1} + {\text{x}}_{2} - 36 \le 0} \hfill \\ {} \hfill & {0 \le {\text{x}}_{1} \le 3,0 \le {\text{x}}_{2} \le 4} \hfill \\ \end{array} $$

The global minimum solution is:

$$ \bar{x} = \left( {2.3295,3.17846} \right),f\left( {\bar{x}} \right) = - 5.50796. $$

Problem 09

$$ \begin{array}{*{20}l} {\text{Minimize}} \hfill & {\text{f} \left( {\text{x}} \right) = 5.3578\,{\text{x}}_{3}^{2} + 0.8357\,{\text{x}}_{1} {\text{x}}_{5} + 37.2392\,{\text{x}}_{1} } \hfill \\ {{\text{Subject}}\,{\text{to:}}} \hfill & {\text{g}_{1} \left( {\text{x}} \right) = 0.00002584\,{\text{x}}_{3} {\text{x}}_{5} - 0.0006663\,{\text{x}}_{2} {\text{x}}_{5} - 0.0000734\,{\text{x}}_{1} {\text{x}}_{4} - 1 \le 0} \hfill \\ {} \hfill & {\text{g}_{2} \left( {\text{x}} \right) = 0.000853007\,{\text{x}}_{2} {\text{x}}_{5} + 0.00009395\,{\text{x}}_{1} {\text{x}}_{4} - 0.00033085\,{\text{x}}_{3} - 1 \le 0} \hfill \\ {} \hfill & {\text{g}_{3} \left( {\text{x}} \right) = 0.00024186\,{\text{x}}_{2} {\text{x}}_{5} + 0.00010159\,{\text{x}}_{1} {\text{x}}_{2} + 0.00007379\,{\text{x}}_{3}^{2} \, - 1 \le 0} \hfill \\ {} \hfill & {\text{g}_{4} \left( x \right) = 1330.3294\,{\text{x}}_{2}^{ - 1} {\text{x}}_{5}^{ - 1} - 0.42\,{\text{x}}_{1} {\text{x}}_{5}^{ - 1} - 0.30586\,{\text{x}}_{2}^{ - 1} {\text{x}}_{3}^{2} {\text{x}}_{5}^{ - 1} - 1 \le 0} \hfill \\ {} \hfill & {\text{g}_{5} \left( {\text{x}} \right) = 2275.1327\,{\text{x}}_{3}^{ - 1} {\text{x}}_{5}^{ - 1} - 0.2668\,{\text{x}}_{1} {\text{x}}_{5}^{ - 1} - 0.40584\,{\text{x}}_{4} {\text{x}}_{5}^{ - 1} - 1 \le 0} \hfill \\ {} \hfill & {\text{g}_{6} \left( {\text{x}} \right) = 0.00029955\,{\text{x}}_{3} {\text{x}}_{5} + 0.00007992\,{\text{x}}_{1} {\text{x}}_{3} + 0.00012157\,{\text{x}}_{3} {\text{x}}_{4} - 1 \le 0} \hfill \\ {} \hfill & {78 \le {\text{x}}_{1} \le 102,33 \le {\text{x}}_{2} \le 45,27 \le {\text{x}}_{\text{i}} \le 45,\,{\text{i}} = 3,4,5.} \hfill \\ \end{array} $$

The global minimum optimal solution is:

$$ \bar{x} = \left( {78.0,33.0,29.998,45.0,36.7673} \right),f\left( {\bar{x}} \right) = 10122.6964. $$

Problem 10

$$ \begin{array}{*{20}l} {{\text{Minimize}} } \hfill & {\text{f} \left( {\text{x}} \right) = \left( {{\text{x}}_{1}^{2} + {\text{x}}_{2} - 11} \right)^{2} + \left( {{\text{x}}_{1} + {\text{x}}_{2}^{2} - 7} \right)^{2} } \hfill \\ {{\text{Subject}}\,{\text{to:}}} \hfill & {\text{g}_{1} \left( {\text{x}} \right) = 4.84 - \left( {{\text{x}}_{1} - 0.05} \right)^{2} - \left( {{\text{x}}_{2} - 2.5} \right)^{2} \ge 0} \hfill \\ {} \hfill & {\text{g}_{2} \left( {\text{x}} \right) = {\text{x}}_{1}^{2} + \left( {{\text{x}}_{2} - 2.5} \right)^{2} - 4.84 \ge 0} \hfill \\ {} \hfill & {0 \le {\text{x}}_{\text{i}} \le 6\,,{\text{i}} = 1,2.} \hfill \\ \end{array} $$

The global minimum optimal solution is:

$$ \bar{x} = \left( {2.246826,2.381865} \right),f\left( {\bar{x}} \right) = 13.59085. $$