Within a nonzero, real Banach space we study the problem of characterising a maximal extension of a monotone operator in terms of minimality properties of representative functions that are bounded by the Penot and Fitzpatrick functions. We single out a property of this space of representative functions that enable a very compact treatment of maximality and pre-maximality issues. This Paper is D...

Sufficient optimality and sensitivity of a parameterized min-max programming with fixed feasible set are analyzed. Based on Clarke’s subdifferential and Chaney’s second-order directional derivative, sufficient optimality of the parameterized min-max programming is discussed first. Moreover, under a convex assumption on the objective function, a subdifferential computation formula of the margina...

It is known that the Karush-Kuhn-Tucker (KKT) conditions of semidefinite programming can be reformulated as a nonsmooth system via the metric projector over the cone of symmetric and positive semidefinite matrices. We show in this paper that the primal and dual constraint nondegeneracies, the strong regularity, the nonsingularity of the B-subdifferential of this nonsmooth system, and the nonsin...

Let f be a Lipschitz mapping of a separable Banach space X to a Banach space Y. We observe that the set of points at which f is diierentiable in a spanning set of directions but not G^ ateaux diierentiable is-directionally porous. Since Borel-directionally porous sets, in addition to being rst category sets, are null in Aronszajn's (or, equivalently, in Gaussian) sense, we obtain an alternative...

This paper is devoted to the study the first-order behavior of the value function of a parametric optimal control problem with linear constraints and a nonconvex cost function. By establishing an abstract result on the Mordukhovich subdifferential of the value function of a parametric mathematical programming problem, we derive a formula for computing the Mordukhovich subdifferential of the val...

In this paper we introduce the concepts of quasimonotone maps and pseudoconvex functions. Moreover, a notion of pseudomonotonicity for multi mappings is introduced; it is shown that, if a function f is continuous, then its pseudoconvexity is equivalent to the pseudomonotonicity of its generalized subdifferential in the sense of Clarke and Rockafellar and prove that a lower semicontinuous functi...

A convex set C ⊆ X∗∗ × X∗ admits the variant Banach-Dieudonné property (VBDP) if the weak∗-strong closure C w×‖·‖ is the smallest set containing C that is closed to all limits of its bounded and weak∗×‖ · ‖ convergent nets. We show in particular, that all convex sets in X∗∗×X∗ admit the VBDP when E∗ := X∗×X∗∗ is weakly-compactly generated (WCG) and hence if E is either a dual separable or a ref...

The approximate subdifferential introduced by Mordukhovich has attracted much attention in recent works on nonsmooth optimization. Potential advantages over other concepts of subdifferentiability might be related to its nonconvexity. This is motivation to study some topological properties more in detail. As the main result, it is shown that any weakly compact subset of any Hilbert space may be ...