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We describe a few applications of semideenite programming in combinatorial optimization. Semideenite programming is a special case of convex programming where the feasible region is an aane subspace of the cone of positive semideenite matrices. There has been much interest in this area lately, partly because of applications in com-binatorial optimization and in control theory and also because o...

The point (0.5, 0.5, 0.5) ∈ P1, i.e. it satisfies the constraints above; however it is not in the convex hull of the matching vectors. The above example motivates the following family of additional constraints (introduced by Edmonds). Observe that for any matching M , the subgraph induced by M on any odd cardinality vertex subset U has at most (|U | − 1)/2 edges. Thus, without losing any of the...

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 [...

In 1863, Michel described a condition characterized by a total absence of differentiated inner ear structures associated with other skull base anomalies, including an abnormal course of the facial nerve and jugular veins. Michel aplasia clearly differs from Michel dysplasia, in which arrest of embryologic development occurs later. Recently, the role of otic capsule formation on mesenchymal diff...