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A hybrid PSO approach for solving non-convex optimization problems

Ganesan T., Vasant P., Elamvazuthy I.

Archives of Control Sciences

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2012

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Vol. 22, no. 1

87-105

EN

The aim of this paper is to propose an improved particle swarm optimization (PSO) procedure for non-convex optimization problems. This approach embeds classical methods which are the Kuhn-Tucker (KT) conditions and the Hessian matrix into the fitness function. This generates a semi-classical PSO algorithm (SPSO). The classical component improves the PSO method in terms of its capacity to search for optimal solutions in non-convex scenarios. In this work, the development and the testing of the refined the SPSO algorithm was carried out. The SPSO algorithm was tested against two engineering design problems which were; ‘optimization of the design of a pressure vessel’ (P1) and the ‘optimization of the design of a tension/compression spring’ (P2). The computational performance of the SPSO algorithm was then compared against the modified particle swarm optimization (PSO) algorithm of previous work on the same engineering problems. Comparative studies and analysis were then carried out based on the optimized results. It was observed that the SPSO provides a better minimum with a higher quality constraint satisfaction as compared to the PSO approach in the previous work.

2

Convergence of Toland's critical points for sequences of D.C. functions and application to the resolution of semilinear elliptic problems

Yassine A., Alaa N., Elhilali Alaoui A.

Control and Cybernetics

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2001

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Vol. 30, no 4

405-417

EN

We prove that if a sequence (fn)n of D.C. functions (Difference of two Convex functions) converges to a D.C. function f in some appropriate way and if un is a critical point of fn, in the sense described by Toland, and is such that (un)n converges to u, then a is a critical point of f, still in Toland's sense. We also build a new algorithm which searches for this critical point u and then apply it in order to compute the solution of a semilinear elliptic equation.

3

Sub-gradient algorithms for computation of extreme eigenvalues of a real symmetric matrix

Yassine A.

Control and Cybernetics

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1998

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Vol. 27, no 3

387-415

EN

The computation of eigenvalues of a matrix is still of importance from both theoretical and practical points of view. This is a significant problem for numerous industrial and scientific situations, notably in dynamics of structures (e.g. Gerardin, 1984), physics (e.g. Rappaz, 1979), chemistry (e.g. Davidson, 1983), economy (e.g. Morishima, 1971; Neumann, 1946), mathematics (e.g. Golub, 1989; Chatelin, 1983, 1984, 1988). The study of eigenvalue problems remains a delicate task, which generally presents numerical difficulties in relation to its sensivity to roundoff errors that may lead to numerical unstabilities, particularly if the eigenvalues are not well separated. In this paper, new subgradient-algorithms for computation of extreme eigenvalues of a symmetric real matrix are presented. Those algorithms are based on stability of Lagrangian duality for non-convex optimization and on duality in the difference of convex functions. Some experimental results which prove the robustness and efficiency of our algorithms are provided.