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Equilibrium programming in Hilbert spaces. (English) Zbl 1109.90079
Given a Hilbert space $\cal{H}$, a closed convex subset $K$ of $\cal{H}$ 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 $a\in\cal{H}$ 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.

90C48Programming in abstract spaces
90C47Minimax problems
49K27Optimal control problems in abstract spaces (optimality conditions)