Hybrid Lagrange multiplier approaches for solving infinite dimensional equilibrium problems with cone constraints
Let \(K\subset B\) be a nonempty closed convex cone of a reflexive Banach space \(B\) and let \(f: K\times K\to\mathbb{R}\) be a smooth function. The authors provide hybrid Lagrange multiplier approaches for equilibrium problems according to NEWLINE\[NEWLINE\text{find an }x^*\in K\text{ such that }f(x^*, y)\geq 0\quad\forall y\in K.NEWLINE\]NEWLINE It is pointed out that many particular problems (e.g. optimization problems, fixed point problems, complementarity problems, variational inequality problems) can be formulated and solved as equilibrium problems.NEWLINENEWLINE The main goal of the paper is the generalization of two finite-dimensional Lagrange multiplier algorithms published in a former paper to the infinite-dimensional case. Both algorithms start with an approximate solution of an unconstrained equilibrium problem. Then, using the so-called Bregman distance with respect to a suitable regularizing function, the next iterations are computed by performing either an extragradient-type step or a Bregman projection onto a certain hyperplane separating the current iterate from the solution set. Under appropriate assumptions, it is shown that the generated sequences converge weakly and globally to a solution of the problem.
scientific article; zbMATH DE number 6039389