On some generalized equations with metrically C-increasing mappings: solvability and error bounds with applications to optimization

Generalized equations are problems emerging in contexts of modern variational analysis as an adequate formalism to treat such issues as constraint systems, optimality and equilibrium conditions, variational inequalities, differential inclusions. The present paper contains a study on solvability and error bounds for generalized equations of the form $F(x)\subseteq C$, where $F$ is a given set-valued mapping and $C$ is a closed, convex cone. A property called metric $C$-increase, matching the metric behaviour of $F$ with the partial order associated with $C$, is singled out, which ensures solution existence and error bound estimates in terms of problem data. Applications to the exact penalization of optimization problems with constraint systems, defined by the above class of generalized equations, and to the existence of ideal efficient solutions in vector optimization are proposed.