On the Fermat-Lagrange principle for mixed smooth convex extremal problems (original) (raw)

Higher–Order Necessary Optimality Conditions for a Nonsmooth Extremum Problem

Journal of Applied Analysis, 2000

By extending the definition of Dini-Hadamard directional derivatives, higher-order necessary optimality conditions for a nonsmooth extremum problem are established, in the presence of both equality and inequality constraints. The presence of a regularity assumption for the inequality constraints strenghtens the optimality condition.

On necessary conditions for infinite-dimensional extremum problems

Journal of Global Optimization, 2004

In this paper, we carry on the analysis (introduced in [4] and developed in [2, 7]) of optimality conditions for extremum problems having infinite-dimensional image, in the case of unilateral constraints. This is done by associating to the feasible set a special multifunction. It turns out that the classic Lagrangian multiplier functions can be factorized into a constant term and a variable one; the former is the gradient of a separating hyperplane as introduced in [4, 5]; the latter plays the role of selector of the above multifunction. Finally, the need of enlarging the class of Lagrangian multiplier functions is discussed.

Optimality conditions in convex optimization revisited

Optimization Letters, 2013

The phrase convex optimization refers to the minimization of a convex function over a convex set. However the feasible convex set need not be always described by convex inequalities. In this article we consider a convex feasible set which are described by inequality constraints which are locally Lipschitz and not necessarily convex and need not be smooth. We show that if the Slater's constraint qualification and a simple non-degeneracy condition is satisfied then the Karush-Kuhn-Tucker type optimality condition is both necessary and sufficient.

Convexity, duality, and Lagrange multipliers

Lecture Notes, MIT Press, …, 2001

These notes were developed for the needs of the 6.291 class at M.I.T. (Spring 2001). They are copyright-protected, but they may be reproduced freely for noncommercial purposes. v vi Preface cation to optimization. For example, in Chapter 1, soon after the development of some of the basic facts about convexity, I discuss some of their applications to optimization and saddle point theory; soon after the discussion of hyperplanes and cones, I discuss conical approximations and necessary conditions for optimality; soon after the discussion of polyhedral convexity, I discuss its application in linear and integer programming; and soon after the discussion of subgradients, I discuss their use in optimality conditions. I follow consistently this style in the remaining chapters, although having developed in Chapter 1 most of the needed convexity theory, the discussion in the subsequent chapters is more heavily weighted towards optimization.

Regularity conditions for constrained extremum problems

Journal of Optimization Theory and Applications, 1985

Necessary and/or sufficient conditions are stated in order to have regularity for nondifferentiable problems or ditterentiable problems. These conditions are compared with some known constraint qualifications. Plenum Publishing Corporation © r Z 9 © Fig, 2, Summary of regularity conditions,

Necessary conditions for an extremum in a mathematical programming problem

Proceedings of the Steklov Institute of Mathematics, 2007

For minimization problems with equality and inequality constraints, first-and second-order necessary conditions for a local extremum are presented. These conditions apply when the constraints do not satisfy the traditional regularity assumptions. The approach is based on the concept of 2-regularity; it unites and generalizes the authors' previous studies based on this concept.

Second-order and related extremality conditions in nonlinear programming

Journal of Optimization Theory and Applications, 1980

This paper is concerned with the problem of characterizing a local minimum of a mathematical programming problem with equality and inequality constraints. The main object is to derive second-order conditions, involving the Hessians of the functions, or related results where some other curvature information is used. The necessary conditions are of the Fritz John type and do not require a constraint qualification. Both the necessary conditions and the sufficient conditions are given in equivalent pairs of primal and dual formulations.