## 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, $$X^{\ast }$$ is its continuous dual, $$K$$ is a convex subset of $$X$$, $$f$$ is a mapping from $$X \times X$$ in the extended reals being 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 $$f( x,y_{x}) <\infty$$ and either $$y_{x}\in \operatorname{int}K$$ or $$y\mapsto f( x,y )$$ is continuous at $$y_{x}$$. An optimization problem
$g( x^{\ast }) \rightarrow \max\tag{D}$ is defined as dual problem for the equilibrium problem $\text{Find }x\in K\text{ such that }f( x,y) \geq 0\;[\geq -\varepsilon]\text{ for all }y\in K\;[\varepsilon\text{-solvability}],\tag{EP}$
where $$g( x^{\ast }) :=\inf_{x\in K}\langle x^{\ast },x\rangle-\inf_{x\in K}\sup_{y\in X}( \langle x^{\ast },y\rangle-f( x,y) )$$ is well defined and non-positive on $$K^{\ast }$$ and $$K^{\ast }$$ is the effektive domain of $$x^*\mapsto\inf_{x\in K}\langle x^* ,x\rangle$$. The following strong duality results are shown:
Theorem 3.1: If $$x$$ is a solution of (EP) then (D) has a solution $$x^{\ast }$$ and $$g( x^{\ast }) =0$$.
Theorem 3.2: (EP) is $$\varepsilon$$-solvable for each $$\varepsilon >0$$ 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 and duality in mathematical programming 49J27 Existence theories for problems in abstract spaces 49J40 Variational inequalities 49J52 Nonsmooth analysis 90C47 Minimax problems in mathematical programming 90C48 Programming in abstract spaces
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### References:

 [7] Hadjisavvas, N. and Schaible, S. (1998), Quasimonotonicity and pseudomonotonicity in variational inequalities and equilibrium problems, in Generalized Convexity, Generalized Monotonicity: Recent Results (Luminy, 1996), Nonconvex Optim. Appl., 27, Kluwer Acad. Publ., Dordrecht, 257–275. · Zbl 0946.49005 [12] Oettli, W. and Schläger, D. (1998), Generalized vectorial equilibrium and generalized monotonicity, in Functional Analysis with Current Applications in Science, Technology and Industry (Aligarh, 1996), Pitman Res. Notes Math. Ser., 377, Longman, Harlow, 145–154. · Zbl 0904.90150
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