Duality for equilibrium problems.

*(English)*Zbl 1106.90074A 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.

\[ g( x^{\ast }) \rightarrow \max\tag{D} \] 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.

Reviewer: Armin Hoffmann (Ilmenau)

##### 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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\textit{J. E. Martínez-Legaz} and \textit{W. Sosa}, J. Glob. Optim. 35, No. 2, 311--319 (2006; Zbl 1106.90074)

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