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Equilibrium programming in Hilbert spaces. (English) Zbl 1109.90079
Given a Hilbert space $ℋ$, a closed convex subset $K$ of $ℋ$ and a countable family of functions ${F}_{i}:{K}^{2}\to R$ ($i\in I$), the authors consider the problem of finding $x\in K$ such that ${F}_{i}\left(x,y\right)\ge 0$ for all $i\in I$ and $y\in K$, as well as the problem of finding the projection of $a\in ℋ$ 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.

##### MSC:
 90C48 Programming in abstract spaces 90C47 Minimax problems 49K27 Optimal control problems in abstract spaces (optimality conditions)