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A league-knock-out tournament quantum particle swarm optimization algorithm for nonlinear constrained optimization problems and applications

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Abstract

In solving optimization problem by metaheuristic algorithm, the main key for avoiding local optima is to enhance the exploration and exploitation phases. This is possible through several processes like enhancement of search agent, hybridization etc. The aim of the work is to introduce a novel hybrid algorithm considering the strategies of group league and knock-out system through advanced metaheuristic algorithm, viz. quantum particle swarm optimization (QPSO). In the tournament of well known games, like Football World Cup, Cricket World Cup, etc. the above average teams (according to their performance) are selected through group league phase in the first round. Then the best team is identified through knock-out phase in the next rounds. This concept is applied in the proposed work to develop a hybrid algorithm for solving non-convex constrained optimization problems using parameter free penalty function technique. In group league phase, five different strategies are considered in developing league-knock-out based hybrid algorithms. For testing the robustness as well as efficiencies of these algorithms, constrained benchmark problems are solved and the results are compared with the existing popular algorithms. The obtained results of different variants are compared statistically and graphically to pick up the best strategy based algorithm. To analyse the statistical significance of the results obtained by hybrid algorithms, three statistical tests (non-parametric) are performed. Finally, this best strategy based hybrid algorithm is applied fruitfully for solving several engineering design problems.

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Acknowledgements

The first author wishes to express his heartfelt gratitude to the University Grants Commission, New Delhi, India, for financial help provided under award No. F. 82-1/2018. (SA-III). Additionally, the fifth author wishes to recognise the financial assistance granted for this study by WBDST&BT, West Bengal, India (Memo No: 429 (Sanc.)/ST/P/S&T/16G-23/2018 dated 12/03/2019). This work is partially supported by the financial assistance of Department of Science and Technology under FIST programme (SR/FST/MSII/2017/10(c) dated 23.10.2018).

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Authors and Affiliations

  1. Department of Mathematics, The University of Burdwan, Purba Burdwan, West Bengal, 713104, India

    Goutam Mandal, Nirmal Kumar, Ali Akbar Shaikh & Asoke Kumar Bhunia

  2. Purulia Polytechnic, P.O.-V. Nagar, Bongabari, Purulia, West Bengal, 723147, India

    Nirmal Kumar

  3. Department of Applied Mathematics, Maulana Abul Kalam Azad University of Technology, WB, Haringhata, Nadia, West Bengal, 741249, India

    Avijit Duary

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  1. Goutam Mandal
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  5. Asoke Kumar Bhunia
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Correspondence to Ali Akbar Shaikh.

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Appendix 1

Appendix 1

P-1: (Levy & Montalvo problem) (Levy and Montalvo 1985).

Minimize \(f(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{v} ) = - v_{1} - v_{2}\).

subject to:

$$\begin{aligned}g_{1} (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{v} )& = [(v_{1} - 1)^{2} + (v_{2} - 1)][1/2\alpha^{2} - 1/2\beta^{2} ]\\&\quad + (v_{1} - 1) + (v_{2} - 1)][1/\alpha^{2} - 1/\beta^{2} ] - 1 \ge 0; \end{aligned}$$

where \(0 \le v_{1} ,v_{2} \le 1\) and \(\alpha = 2\), \(\beta = 0.25.\)

The global minimum of this problem is given by \(f_{\min } = - 1.8729\) with \(v_{1} = 1,v_{2} = 0.8729.\)


P-2: (Himmelblau problem) (Himmelblau 2018).

Minimize \(f(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{v} ) = 4.3v_{1} + 31.8v_{2} + 63.3v_{3} + 15.8v_{4} + 68.3v_{5} + 4.7v_{6}\).

subject to

$$\begin{gathered} g_{1} (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{v} ) = 17.1v_{1} + 38.2v_{2} + 204.2v_{3} + 212.3v_{4} + 623.4v_{5} + 1495.5v_{6} - 169v_{1} v_{3} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - 3580v_{3} v_{5} - 3810v_{4} v_{5} - 18500v_{4} v_{6} - 24300v_{5} v_{6} - 4.97 \ge 0; \hfill \\ \end{gathered}$$
$$\begin{gathered} g_{2} (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{v} ) = 1.88 + 17.9v_{1} + 36.8v_{2} + 133.9v_{3} + 169.7v_{4} + 337.8v_{5} + 1385.2v_{6} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - 139v_{1} v_{3} - 2450v_{4} v_{5} - 600v_{4} v_{6} - 17200v_{5} v_{6} \ge 0; \hfill \\ \end{gathered}$$
$$g_{3} (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{v} ) = 429.08 - 273v_{2} - 70v_{4} - 819v_{5} + 26000v_{4} v_{5} \ge 0;$$
$$g_{4} (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{v} ) = 159.9v_{1} - 311v_{2} + 587v_{4} + 391v_{5} + 2198v_{6} - 14000v_{1} v_{6} + 78.02 \ge 0;$$

where \(0 \le v_{1} \le 0.31,\)\(0 \le v_{2} \le 0.046,\)\(0 \le v_{3} \le 0.068,\)\(0 \le v_{4} \le 0.042,\)\(0 \le v_{5} \le 0.028,\)\(0 \le v_{6} \le 0.0134.\)

The global minimum of this problem occurs at (0, 0, 0, 0, 0, 0.00333) with \(f_{\min } = 0.0156\).


P-3: (Hess problem) (Hesse 1973).

Maximize \(f(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{v} ) = 25(v_{1} - 2)^{2} + (v_{2} - 2)^{2} + (v_{3} - 1)^{2} + (v_{4} - 4)^{2} + (v_{5} - 1)^{2} + (v_{6} - 4)^{2}\).


subject to

\(g_{1} (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{v} ) = v_{1} + v_{2} - 2.0 \ge 0\);

$$g_{2} (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{v} ) = - v_{1} + v_{2} + 6.0 \ge 0;$$

\(g_{3} (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{v} ) = v_{1} - v_{2} + 2.0 \ge 0\);

\(g_{4} (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{v} ) = - v_{1} + 3v_{2} + 2.0 \ge 0\);

\(g_{5} (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{v} ) = (v_{3} - 3)^{2} + v_{4} - 4 \ge 0\);

$$g_{6} (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{v} ) = (v_{5} - 3)^{2} + v_{6} - 4 \ge 0;$$

The ranges of the variables are \(0 \le v_{1} \le 5,\)\(0 \le v_{2} \le 1,\)\(0 \le v_{3} \le 5,\)\(0 \le v_{4} \le 6,\)\(0 \le v_{5} \le 5,\)\(0 \le v_{6} \le 10\). The function attains minimum value at (5, 1, 5, 0, 5, 10) with minimum value 310.


P-4: (Schittkowski problem) (Schittkowski 1987).

Minimize \(f(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{v} ) = (v_{1}^{2} + v_{2} - 11)^{2} + (v_{1} + v_{2}^{2} - 7)^{2}\).

Subject to

$$\begin{gathered} g_{1} (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{v} ) = 4.84 - (v_{1} - 0.05)^{2} - (v_{2} - 2.5)^{2} \ge 0; \hfill \\ g_{2} (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{v} ) = v_{1}^{2} + (v_{2} - 2.5)^{2} - 4.84 \ge 0; \hfill \\ \end{gathered}$$

with \(0 \le v_{j} \le 6,\)\(j = 1,2\)

Here \(f_{\min }\) = 13.59085 at \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{v}^{*} = (2.246826,2.381865)\).


P-5: (Schoenauer & Xanthakis problem) (Schoenauer and Xanthakis 1993).

Maximize \(f(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{v} ) = \frac{{\sin^{3} (2\pi v_{1} )\sin (2\pi v_{2} )}}{{v_{1}^{3} (v_{1} + v_{2} )}}\).

subject to

$$\begin{gathered} g_{1} (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{v} ) = - v_{1}^{2} + v_{2} - 1 \ge 0 \hfill \\ g_{2} (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{v} ) = - 1 + v_{1} - (v_{2} - 4)^{2} \ge 0 \hfill \\ \end{gathered}$$

where \(0 \le v_{j} \le 10,\)\(j = 1,2\).

The global maximum of this function is at (1.2279713, 4.2453733) with \(f_{\max } = 0.095\).


P-6: (Salkin problem) (Salkin 1975).

Maximize \(f(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{v} ) = 3v_{1} + v_{2} + 2v_{3} + v_{4} - v_{5}\).

subject to

\(g_{1} (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{v} ) = 25v_{1} - 40v_{2} + 16v_{3} + 21v_{4} + v_{5} \le 300\);

\(g_{2} (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{v} ) = v_{1} + 20v_{2} - 50v_{3} + v_{4} - v_{5} \le 200\);

\(g_{3} (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{v} ) = 60v_{1} + v_{2} - v_{3} + 2v_{4} + v_{5} \le 600\);

$$g_{4} (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{v} ) = - 7v_{1} + 4v_{2} + 15v_{3} - v_{4} + 65v_{5} \le 700;$$

with \(1 \le v_{1} \le 4,\)\(80 \le v_{2} \le 88,\)\(30 \le v_{3} \le 35,\)\(145 \le v_{4} \le 150,\)\(0 \le v_{5} \le 2\).

Here \(f_{\max }\) = 320 at \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{v}^* = (4,88,35,150,0)\).


P-7: (Michalewicz problem) (Zbigniew 1996).

Minimize \(f(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{v} ) = 6.5 v_{1} - 0.5v_{1}^{2} - v_{2} - 2v_{3} - 3v_{4} - 2v_{5} - v_{6}\).

subject to

$$\begin{gathered} g_{1} (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{v} ) = v_{1} + 2v_{2} + 8v_{3} + v_{4} + 3v_{5} + 5v_{6} \le 16; \hfill \\ g_{2} (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{v} ) = - 8v_{1} - 4v_{2} - 2v_{3} + 2v_{4} + 4v_{5} - v_{6} \le - 1; \hfill \\ g_{3} (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{v} ) = 2v_{1} + 0.5v_{2} + 0.2v_{3} - 3v_{4} - v_{5} - 4v_{6} \le 24; \hfill \\ g_{4} (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{v} ) = 0.2v_{1} + 2v_{2} + 0.1v_{3} - 4v_{4} + 2v_{5} + 2v_{6} \le 12; \hfill \\ g_{5} (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{v} ) = - 0.1v_{1} - 0.5v_{2} + 2v_{3} + 5v_{4} - 5v_{5} + 3v_{6} \le 3; \hfill \\ \end{gathered}$$

where \(0 \le v_{j} \le 2,\)\(j = 1,3,6\),\(0 \le v_{2} \le 10,\)\(0 \le v_{k} \le 1,\)\(k = 4,5\)

Here \(f_{\min }\)\(= - 11\) at \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{v}^* = (0,6,0,1,1,0)\).


P-8: (Michaelwicz & Schoenauer problem) (Michalewicz and Schoenauer 1996).

Minimize \(f(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{v} ) = 5\sum\limits_{i = 1}^{4} {v_{i} - } 5\sum\limits_{i = 1}^{4} {v_{{_{i} }}^{2} - } \sum\limits_{i = 5}^{13} {v_{i} }\).

subject to

$$\begin{gathered} g_{1} (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{v} ) = 10 - (2v_{1} + 2v_{2} + v_{10} + v_{11} ) \ge 0; \hfill \\ g_{2} (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{v} ) = 10 - (2v_{1} + 2v_{3} + v_{10} + v_{12} ) \ge 0; \hfill \\ g_{3} (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{v} ) = 10 - (2v_{2} + 2v_{3} + v_{11} + v_{12} ) \ge 0; \hfill \\ g_{4} (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{v} ) = 8v_{1} - v_{10} \ge 0; \hfill \\ g_{5} (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{v} ) = 8v_{2} - v_{11} \ge 0; \hfill \\ \end{gathered}$$
$$\begin{gathered} g_{6} (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{v} ) = 8v_{3} - v_{12} \ge 0; \hfill \\ g_{7} (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{v} ) = 2v_{4} + v_{5} - v_{10} \ge 0; \hfill \\ g_{8} (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{v} ) = 2v_{6} + v_{7} - v_{11} \ge 0; \hfill \\ g_{9} (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{v} ) = 2v_{8} + v_{9} - v_{12} \ge 0; \hfill \\ \end{gathered}$$

where \(0 \le v_{j} \le 1,\)\(j = 1,2,3,4,5,6,7,8,9,13\), \(0 \le v_{k} \le 100\), \(k = 10,11,12\).

Here \(f_{\min }\)\(= - 15\) at \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{v}^* = (1,1,1,1,1,1,1,1,1,3,3,3,1)\).

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Mandal, G., Kumar, N., Duary, A. et al. A league-knock-out tournament quantum particle swarm optimization algorithm for nonlinear constrained optimization problems and applications. Evolving Systems 14, 1117–1143 (2023). https://doi.org/10.1007/s12530-023-09485-1

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