Equilibrium programming in Hilbert spaces. (English) Zbl 1109.90079

Combettes, Patrick L.; Hirstoaga, Sever A.

J. Nonlinear Convex Anal. 6, No. 1, 117-136 (2005).

Given a Hilbert space \(\mathcal{H}\), a closed convex subset \(K\) of \(\mathcal{H}\) and a countable family of functions \(F_{i}\colon K^2\to R\) (\(i\in I\)), the authors consider the problem of finding \(x\in K\) such that \(F_{i}(x,y)\geq0\) for all \(i\in I\) and \(y\in K\), as well as the problem of finding the projection of \(a\in\mathcal{H}\) on \(S\), the solution set of the preceding problem. In order to accomplish these aims, proximal-like block-iterative algorithms, as well as regularization and splitting algorithms, are proposed. For every algorithm, convergence results are established.

Reviewer: Constantin Zălinescu (Iaşi) (MR 2006a:90151)

MSC:

90C48	Programming in abstract spaces
90C47	Minimax problems in mathematical programming
49K27	Optimality conditions for problems in abstract spaces