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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 xK there is a y x K such that f(x,y x )< and either y x intK or yf(x,y) is continuous at y x . An optimization problem

g(x * )max(D)

is defined as dual problem for the equilibrium problem

FindxKsuchthatf(x,y)0[-ε]forallyK[ε-solvability],( EP )

where g(x * ):=inf xK x * ,x-inf xK sup yX (x * ,y-f(x,y)) is well defined and non-positive on K * and K * is the effektive domain of x * inf xK 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(x * )=0.

Theorem 3.2: (EP) is ε-solvable for each ε>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:
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
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