Jean-Baptiste Hiriart-Urruty

Fundamentals of convex analysis

Introduction: Notation, Elementary Results.- Convex Sets: Generalities Convex Sets Attached to a Convex Set Projection onto Closed Convex Sets Separation and Applications Conical Approximations of Convex Sets.- Convex Functions: Basic Definitions and Examples Functional Operations Preserving Convexity Local and Global Behaviour of a Convex Function First- and Second-Order Differentiation.- Sublinearity and Support Functions: Sublinear Functions The Support Function of a Nonempty Set Correspondence Between Convex Sets and Sublinear Functions.- Subdifferentials of Finite Convex Functions: The Subdifferential: Definitions and Interpretations Local Properties of the Subdifferential First Examples Calculus Rules with Subdifferentials Further Examples The Subdifferential as a Multifunction.- Conjugacy in Convex Analysis: The Convex Conjugate of a Function Calculus Rules on the Conjugacy Operation Various Examples Differentiability of a Conjugate Function.

Tangent Cones, Generalized Gradients and Mathematical Programming in Banach Spaces

This study is devoted to constrained optimization problems in Banach spaces. We present different properties of tangent cones associated with an arbitrary subset of a Banach space and establish correlations with some of the existing results. In absence of both differentiability and convexity assumptions on the functions involved in the optimization problem, the consideration of these tangent cones and their polars leads us to introduce new concepts in nondifferentiable programming. Necessary optimality conditions are first developed in a general abstract form; then these conditions are made more precise in the presence of equality constraints by introducing the concept of normal subcone.

Generalized Hessian matrix and second-order optimality conditions for problems withC1,1 data

In this paper, we present a generalization of the Hessian matrix toC1,1 functions, i.e., to functions whose gradient mapping is locally Lipschitz. This type of function arises quite naturally in nonlinear analysis and optimization. First the properties of the generalized Hessian matrix are investigated and then some calculus rules are given. In particular, a second-order Taylor expansion of aC1,1 function is derived. This allows us to get second-order optimality conditions for nonlinearly constrained mathematical programming problems withC1,1 data.

On optimality conditions in nondifferentiable programming

This paper is devoted to necessary optimality conditions in a mathematical programming problem without differentiability or convexity assumptions on the data. The main tool of this study is the concept of generalized gradient of a locally Lipschitz function (and more generally of a lower semi-continuous function). In the first part, we consider local extremization problems in the unconstrained case for objective functions taking values in (−∞, +∞]. In the second part, the constrained case is considered by the way of the cone of adherent displacements. In the presence of inequality constraints, we derive in the third part optimality conditions in the Kuhn—Tucker form under a constraint qualification.

A Variational Approach to Copositive Matrices

This work surveys essential properties of the so-called copositive matrices, the study of which has been spread over more than fifty-five years. Special emphasis is given to variational aspects related to the concept of copositivity. In addition, some new results on the geometry of the cone of copositive matrices are presented here for the first time.

Conditions for Global Optimality 2

In this paper bearing the same title as our earlier survey-paper [11] we pursue the goal of characterizing the global solutions of an optimization problem, i.e. getting at necessary and sufficient conditions for a feasible point to be a global minimizer (or maximizer) of the objective function. We emphasize nonconvex optimization problems presenting some specific structures like ‘convex-anticonvex’ ones or quadratic ones.

Global Optimality Conditions in Maximizing a Convex Quadratic Function under Convex Quadratic Constraints

For the problem of maximizing a convex quadratic function under convex quadratic constraints, we derive conditions characterizing a globally optimal solution. The method consists in exploiting the global optimality conditions, expressed in terms of ∈-subdifferentials of convex functions and ∈-normal directions, to convex sets. By specializing the problem of maximizing a convex function over a convex set, we find explicit conditions for optimality.

What is the subdifferential of the closed convex hull of a function

Given a function

A general formula on the conjugate of the difference of functions

f:\mathbb{R}^n \to ( { - \infty , + \infty } ]

Extension of Lipschitz functions

and its closed convex hull