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

Abstract

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) ⊆ 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.

A condition for the stability of ideal efficient solutions in parametric vector optimization via set-valued inclusions

“…Throughout the current section, (X, d) stands for a complete metric space, whereas (Y, • ) denotes a real Banach space. Such existence condition refines and accomplishes an analogous result recently proposed (see [27,Theorem 5.1]), by weakening several of its hypotheses. Indeed, the continuity of f is replaced by the lower C-semicontinuity, while the closedness of f (R) is dropped out.…”

“…Indeed, the continuity of f is replaced by the lower C-semicontinuity, while the closedness of f (R) is dropped out. Besides, an assumption, given for granted in [27,Theorem 5.1], is now explicitly made, which avoids a pathological, yet possible, behaviour of (VOP).…”

“…The quantity inc(g; S) = sup{a > 1 : inclusion (6) holds} is called the exact bound of metric C-increase of g on S. For a discussion of this notion, including examples, related properties and its connection with the decrease principle of variational analysis, the reader is referred to [27].…”

“…function, it is closed. Now, in order to show that IE = ∅ and the error bound in (7) holds true, it suffices to apply [27,Theorem 4.2], with X = R, F = f (R) − f and φ = ν, following the same argument as proposed in [27,Theorem 5.1]. In doing so, notice that the existence of x 0 ∈ R such that ν(x 0 ) < +∞ is guaranteed by hypothesis (ii), whereas the lower semicontinuity of ν can be derived directly from the lower C-semicontinuity of f , instead of from the lower semicontinuity of F. Besides, the hypothesis (iv) entails the property of metric C-increase on R for the mapping f (R) − f .…”

“…Set-valued inclusions, in simple as well as in parameterized form, have been recently studied from several viewpoints in [3,27,28,29,30]. The idea underlying the research exposed in the present paper is that useful insights into the stability behaviour of ideal efficient solutions can be obtained by refining in a proper way the study of solution stability of parameterized set-valued inclusions.…”