Dual approach for a class of implicit convex optimization problems (Q706374)

Considered is the class of implicit optimization problems with nonmonotone perturbations i.e. the problems of finding a pair of points \(( x^{\ast },y^{\ast }) \in X\times Y\) such that \[ f( x^{\ast }) +\langle y^{\ast },h( x^{\ast }) \rangle \leq f( x) +\langle y^{\ast },h( x) \rangle \;\;\forall x\in X,\eqno(1) \] \[ \langle b( y^{\ast }) -h( x^{\ast }) ,y-y^{\ast }\rangle \geq 0\;\;\forall y\in Y\eqno(2) \] where \(X\) is a nonempty, convex, and closed subset of the \(n\)-dimensional Euclidean space \(\mathbb{R}^{n};\;h:X\to \mathbb{R}^{m}\) a continuous mapping with convex continuous components \(h_{i}:X\to \mathbb{R}\) for \(i=1,\dots ,m;\) \( f:X\to \mathbb{R}\) a convex continuous function and \(Y\) is a nonempty, convex and closed subset of \(\mathbb{R}_{+}^{m}\). The author suggests the problem can be converted into a mixed variational inequality formulation of optimality conditions for a nonconvex and nonsmooth optimization pronlem which can be solved by splitting type methods. In the last section he considers some additional examples of equilibrium type problems which can be viewed as particular cases of the system (1),(2).