The relationships between the generalized directional derivative of the distance function and the existence of nearest points as well as some geometry properties in Banach spaces are studied. It is proved in the present paper that the condition that for each closed subset G of X and x ∈ X \ G, the Clarke, Michel-Penot, Dini or modified Dini directional derivative of the distance function is 1 o...

In this paper we study the mathematical program with geometric constraints such that the image of a mapping from a Banach space is included in a nonempty and closed subset of a finite dimensional space. We obtain the nonsmooth enhanced Fritz John necessary optimality conditions in terms of the approximate subdifferential. In the case where the Banach space is a weakly compactly generated Asplun...

We apply nonsmooth analysis to a well known optical inverse problem, phase retrieval. The phase retrieval problem arises in many different modalities of electromagnetic imaging and has been studied in the optics literature for over forty years. The state of the art for this problem in two dimensions involves iterated projections for solving a nonconvex feasibility problem. Despite widespread us...

Our approach to the Karush-Kuhn-Tucker theorem in [OSC] was entirely based on subdifferential calculus (essentially, it was an outgrowth of the two subdifferential calculus rules contained in the Fenchel-Moreau and Dubovitskii-Milyutin theorems, i.e., Theorems 2.9 and 2.17 of [OSC]). On the other hand, Proposition B.4(v) in [OSC] gives an intimate connection between the subdifferential of a fun...

Monotone operators are of basic importance in optimization as they generalize simultaneously subdifferential operators of convex functions and positive semidefinite (not necessarily symmetric) matrices. In 1970, Asplund studied the additive decomposition of a maximal monotone operator as the sum of a subdifferential operator and an “irreducible” monotone operator. In 2007, Borwein and Wiersma [...

We present the technical background material for a version of the Pontryagin Maximum Principle with state space constraints and very weak technical hypotheses, based on a primal approach that uses generalized differentials and packets of needle variations. In particular, we give a detailed account of two theories of generalized differentials, the " generalized differential quotients " (GDQs) an...

The classical Jensen inequality for concave function φ is adapted for the Sugeno integral using the notion of the subdifferential. Some examples in the framework of the Lebesgue measure to illustrate the results are presented.