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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

2001

Vol. 30, no 4

405-417

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.

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

Yassine A.

Control and Cybernetics

1998

Vol. 27, no 3

387-415

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.