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Interior proximal method for variational inequalities on non-polyhedral sets

100%

Kaplan A. , Tichatschke R.

tom 27

nr 1

71-93

Interior proximal methods for variational inequalities are, in fact, designed to handle problems on polyhedral convex sets or balls, only. Using a slightly modified concept of Bregman functions, we suggest an interior proximal method for solving variational inequalities (with maximal monotone operators) on convex, in general non-polyhedral sets, including in particular the case in which the set is described by a system of linear as well as strictly convex constraints. The convergence analysis of the method studied admits the use of the 𝝐-enlargement of the operator and an inexact solution of the subproblems.

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

51%

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

Control and Cybernetics

2001

tom 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.