Jean-Paul Penot

Characterizations of generalized convexities via generalized directional derivatives

The paper gives characterizations of convexity, quasiconvexity, invexity and pseudoconvexity for a (radially) upper-semicontinuous function f in a topological vector space via appropriate properties of a bifunction which is majorized by the upper radial derivative of f and which stands for a generalized derivative of some sort.

Calmness and Stability Properties of Marginal and Performance Functions

Abstract Semicontinuity properties, calmness properties and Lipschitz type properties of value functions of parametrized optimization problems are examined. No special structure on the decision space is assumed. In particular, tameness conditions in the sense of Rockafellar are out of reach in such a bare framework. Our methods rely on a study of the behavior of the approximate solution sets.

Open mappings theorems and linearization stability

An important computation rule for tangent cones is examined. Two results are given which assume only Hadamard differentiability (and a variant of it) instead of strict Frechet differentiability. This allows the consideration of concrete examples such as superposition operators and can be applied to the problem of linearizing a nonlinear equation or inequality.

Projective dualities for quasiconvex problems

We study two dualities that can be applied to quasiconvex problems. They are conjugacies deduced from polarities. They are characterized by the polar sets of sublevel sets. We give some calculus rules for the associated subdifferentials and we relate the subdifferentials to known subdifferentials. We adapt the general duality schemes in terms of Lagrangians or in terms of perturbations to two specific problems. First a general mathematical programming problem and then a programming problem with linear constraints.

Revisiting the problem of integrability in utility theory

We revisit the problem of integrability in the consumer theory, focusing on the main difficulties. First, we look for a neat and simple local existence result, and then for a global solution. Second, observing that a utility function (or indirect utility function) cannot be determined uniquely, we propose a means to get a kind of uniqueness result. Our approach is coordinate-free and can be used both in the classical case of a finite-dimensional commodity space and in the case an infinite-dimensional model is adopted.

Representation of generalized monotone multimaps

Abstract We survey the role of generalized dualities when dealing with generalized monotone operators, observing that for many conjugacies the coupling function is neither bilinear nor finitely valued. We also make a comparison with the use of bifunctions considered in a similar perspective. We introduce a class of operators close to the class of accretive operators and we raise some open questions.

Subdifferentiation of integral functionals

We examine how the subdifferentials of nonconvex integral functionals can be deduced from the subdifferentials of the corresponding integrand or at least be estimated with the help of them. In fact, assuming some regularity properties of the integrands, we obtain exact expressions for the subdifferentials of the integral functionals. We draw some consequences in terms of duality for such integral functionals, extending in this way the early work of Rockafellar to the nonconvex case.

A note on the connection between Chaney’s derivatives and epi-derivatives

An example is provided showing the necessity of a finiteness assumption in a result of the second author ensuring that the second-order Chaney derivative coincides with the second-order Rockafellar epi-derivative of a lower semicontinuous function.

Optimality Conditions in Semidefinite Programming

Semidefinite positiveness of operators on Euclidean spaces is characterized. Using this characterization, we compute in a direct way the first-order and second-order tangent sets to the cone of semidefinite positive operators on such a space. These characterizations are useful for optimality conditions in semidefinite programming.