Equilibrium programming in Hilbert spaces. (English) Zbl 1109.90079

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.


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