Generalized differentials of nonsmooth functions, and necessary conditions for an extremum (original) (raw)

Generalized differentials of nonsmooth functions

1981

In this article we introduce an abstract notion of extremeness, which similarly to the corresponding definition of Neustadt [I, 2] embraces different notions of a solution to extremal problems, and establish necessary conditions for extremeness.

Generalized convex functions and generalized differentials

Optimization, 2013

We study some classes of generalized convex functions, using a generalized di¤erential approach. By this we mean a set-valued mapping which stands either for a derivative, a subdi¤erential or a pseudo-di¤erential in the sense of Jeyakumar and Luc. We establish some links between the corresponding classes of pseudoconvex, quasiconvex and another class of generalized convex functions we introduced. We devise some optimality conditions for constrained optimization problems. In particular, we get Lagrange-Kuhn-Tucker multipliers for mathematical programming problems.

First-order optimality conditions in set-valued optimization

Mathematical Methods of Operations Research, 2006

A a set-valued optimization problem min C F (x), x ∈ X 0 , is considered, where X 0 ⊂ X, X and Y are Banach spaces, F : X 0 Y is a set-valued function and C ⊂ Y is a closed cone. The solutions of the set-valued problem are defined as pairs (x 0 , y 0 ), y 0 ∈ F (x 0 ), and are called minimizers. In particular the notions of w-minimizer (weakly efficient points), p-minimizer (properly efficient points) and i-minimizer (isolated minimizers) are introduced and their characterization in terms of the so called oriented distance is given. The relation between p-minimizers and i-minimizers under Lipschitz type conditions is investigated. The main purpose of the paper is to derive first order conditions, that is conditions in terms of suitable first order derivatives of F , for a pair (x 0 , y 0 ), where x 0 ∈ X 0 , y 0 ∈ F (x 0 ), to be a solution of this problem. We define and apply for this purpose the directional Dini derivative. Necessary conditions and sufficient conditions a pair (x 0 , y 0 ) to be a w-minimizer, and similarly to be a i-minimizer are obtained. The role of the i-minimizers, which seems to be a new concept in set-valued optimization, is underlined. For the case of w-minimizers some comparison with existing results is done.

Karush–Kuhn–Tucker Conditions for Nonsmooth Mathematical Programming Problems in Function Spaces

SIAM Journal on Control and Optimization, 2011

Lagrange multiplier rules for abstract optimization problems with mixed smooth and convex terms in the cost, with smooth equality constrained and convex inequality constraints are presented. The typical case for the equality constraints that the theory is meant for is given by differential equations. Applications are given to L 1-minimum norm control problems, L ∞norm minimization, and a class of optimal control problems with distributed state constraints and non-smooth cost.

On the degree and separability of nonconvexity and applications to optimization problems

Mathematical Programming, 1997

We study qualitative indications for d.c. representations of closed sets in and functions on Hilbert spaces. The first indication is an index of nonconvexity which can be regarded as a measure for the degree of nonconvexity. We show that a closed set is weakly closed if this indication is finite. Using this result we can prove the solvability of nonconvex minimization problems. By duality a minimization problem on a feasible set in which this indication is low, can be reduced to a quasi-concave minimization over a convex set in a low-dimensional space. The second indication is the separability which can be incorporated in solving dual problems. Both the index of nonconvexity and the separability can be characteristics to "good" d,c. representations. For practical computation we present a notion of clouds which enables us to obtain a good d.c. representation for a class of nonconvex sets. Using a generalized Caratheodory's theorem we present various applications of clouds.

Derivative Characterization of Constrained Extrema of Functionals a Survey

One of the basic axioms of economic analysis is the axiom of rationality, i.e., of postulating that economic behavior results from a process of optimization. As it is, at best, di¢ cult to directly test the axiom, economists have been interested in characterization theorems of privatization theory. In this paper we review the theorems that characterize optimality by way of derivatives. First we formulate a very general optimization problem. Then we present characterization theorems for three types of problems: Finite dimensional, variation and problems in linear topological spaces. In each case we present theorems for equalityinequality constraints. The theorems in each case are: …rst order necessary conditions, …rst order su¢ cient conditions, second order necessary conditions and second order su¢ cient conditions. The scheme of representation is as follows: Statement of theorems are followed by remarks referring the reader to the earliest, known to us, proofs of the theorems. In some instances, slight generalizations of some theorems appear here for the …rst time, an indication of necessary modi…cation to existing proofs are indicated. A case is "solved" if proofs for all the four types of characterization theorems exist. The only "unsolved" case