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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, $X^{\ast }$ is its continuous dual, $K$ is a convex subset of $X$, $f$ is a mapping from $X \times X$ 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( 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.

90C46Optimality conditions, duality
49J27Optimal control problems in abstract spaces (existence)
49J40Variational methods including variational inequalities
49J52Nonsmooth analysis (other weak concepts of optimality)
90C47Minimax problems
90C48Programming in abstract spaces
Full Text: DOI
[1] · Zbl 0383.49005 · doi:10.1016/0022-247X(77)90222-0
[2] · Zbl 0878.49007 · doi:10.1023/A:1022603406244
[3] · Zbl 0903.49006 · doi:10.1007/BF02192244
[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
[8] · Zbl 1017.49008 · doi:10.1016/S0362-546X(02)00154-2
[9] · Zbl 1016.90066 · doi:10.1023/A:1004665830923
[10] · Zbl 0773.90092 · doi:10.1016/0362-546X(92)90159-C
[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