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Duality for equilibrium problems. (English) Zbl 1106.90074

A new duality formulation (D) for general equilibrium (EP) problems is proposed on the basis of convex optimization in locally convex vector spaces (LCVS). Assume that $X$ is a LCVS, ${X}^{*}$ is its continuous dual, $K$ is a convex subset of $X$, $f$ is a mapping from $X×X$ in the extended reals beeing convex and lower semicontinuous in the second variable. Further assume that for all $x\in K$ there is a ${y}_{x}\in K$ such that $f\left(x,{y}_{x}\right)<\infty$ and either ${y}_{x}\in intK$ or $y↦f\left(x,y\right)$ is continuous at ${y}_{x}$. An optimization problem

$g\left({x}^{*}\right)\to max\phantom{\rule{2.em}{0ex}}\left(\mathrm{D}\right)$

is defined as dual problem for the equilibrium problem

$\text{Find}\phantom{\rule{4.pt}{0ex}}x\in K\phantom{\rule{4.pt}{0ex}}\text{such}\phantom{\rule{4.pt}{0ex}}\text{that}\phantom{\rule{4.pt}{0ex}}f\left(x,y\right)\ge 0\phantom{\rule{4pt}{0ex}}\left[\ge -\epsilon \right]\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{all}\phantom{\rule{4.pt}{0ex}}y\in K\phantom{\rule{4pt}{0ex}}\left[\epsilon \text{-solvability}\right],\phantom{\rule{2.em}{0ex}}\left(\mathrm{EP}\right)$

where $g\left({x}^{*}\right):={inf}_{x\in K}〈{x}^{*},x〉-{inf}_{x\in K}{sup}_{y\in X}\left(〈{x}^{*},y〉-f\left(x,y\right)\right)$ is well defined and non-positive on ${K}^{*}$ and ${K}^{*}$ is the effektive domain of ${x}^{*}↦{inf}_{x\in K}〈{x}^{*},x〉$. The following strong duality results are shown:

Theorem 3.1: If $x$ is a solution of (EP) then (D) has a solution ${x}^{*}$ and $g\left({x}^{*}\right)=0$.

Theorem 3.2: (EP) is $\epsilon$-solvable for each $\epsilon >0$ if and only if the optimal value of (D) is zero. It is proposed to solve (D) instead of (EP). Applications to quadratic convex-concave saddle point problems and general convex optimization problems are given.

##### MSC:
 90C46 Optimality conditions, duality 49J27 Optimal control problems in abstract spaces (existence) 49J40 Variational methods including variational inequalities 49J52 Nonsmooth analysis (other weak concepts of optimality) 90C47 Minimax problems 90C48 Programming in abstract spaces
##### References:
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