Jean-Paul Penot

Sorbonne University | UPMC · Institut de mathématiques de Jussieu (IMJ - UMR 7586)

We propose a definition of Lipschizian manifold that is more precise than the notion of Lipschitzian parameterization. It is modelled on the notion of differentiable manifold. We also give a notion of Lipschitzian submanifold and compare it with a notion devised by R.T. Rockafellar (Ann. I.H.P. Sect. C 2(3), 167–184, 1985). Among the examples we me...

We introduce variants of the properties mentioned in the title that bring some versatility. Our approach is geometric but can be related to the more analytical tools considered by A.D. Ioffe. An equivalence result is proved with such a formalism. This general framework is shown to encompass various properties such as subregularity and calmness.

We survey some processes that relate a given function to a more regular function. We examine the compensated convexity process from this point of view and we give a special attention to an infimal convolution approximation generalizing the Moreau approximation which can be applied to nonconvex functions satisfying mild growth conditions.

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...

Convex analysis is devoted to the study and the use of four notions: conjugate functions, normal cones, subdifferentials, support functions under convexity assumptions. These notions are closely related and the calculus rules for one of them imply calculus rules for the other ones. But for none of these notions such rules are always valid without a...

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 c...

We present a simple approach to an analysis of higher order approximations to sets and functions. The objects we study are not of a specific order; they include objects of order 2 and m with m not necessarily an integer. We deduce from these concepts optimality conditions of higher order and we establish some calculus rules.

Using the notions of measure theory introduced in Chap. 1, an integration process is introduced for functions with values in normed vector spaces. Such an extension does not require much supplementary effort but can be bypassed in a first reading. Convergence results and calculus rules form the bulk of the chapter.

This chapter is devoted to a rather complete exposition of classical differential calculus. However, we present some non-classical variants which are often easier to handle. Besides the idea of approximation carried by the notion of derivative, important existence theorems can be obtained: the inverse function theorem and its relative, the implicit...

Hilbert spaces form a major class of normed spaces. They offer geometric properties that are similar to those of Euclidean spaces. In particular one can identify them with their duals and, given a nonempty closed convex subset of such a space, to every point of the space corresponds a closest point in the set. When the set is a linear subspace, thi...

In this chapter, problems involving time are considered. Those expressed by means of ordinary differential equations are the simplest ones. In contrast to problems involving partial derivatives, they do not require the functions spaces introduced in the preceding chapter. But for parabolic problems and hyperbolic problems, Sobolev spaces are again...

A large part of this chapter is devoted to Sobolev spaces, which are convenient spaces for handling partial differential equations. The weakened notion of derivative they convey is related to the question of transposition. Such a notion gives a natural approach to the concept of a weak solution to a partial differential equation. The question of re...

The aim of this chapter is twofold. In the first part vectorial measures are introduced and the question of the representation of a measure in terms of another one is tackled. In the second part, Lebesgue L p spaces are studied and their main properties established. The main properties of the Fourier transform and the Radon transform are displayed...

The notion of limit is central in analysis. Thus the concept of convergence is presented in a general framework and then in the classes of topological spaces and metric spaces. Compactness, connectedness, completeness are studied in detail. Baire’s Theorem is included as well as Ekeland’s Variational Principle. The contraction theorem is proved and...

This chapter can be conceived as a substantial course on convex analysis. But it appears here in view of its relationships with other subjects such as optimization and differential calculus. Convex functions have remarkable continuity and differentiability properties. They offer a substitute to the derivative, the subdifferential, whose calculus ru...

This chapter is devoted to some preliminary subjects and techniques. Sets and orders are briefly considered, in particular for defining nets and sequences which are used throughout the book. Basic facts about countability are reviewed. Since a practice in set theory is needed for topology and analysis, measure spaces are chosen for training. The cl...

In this central chapter, the fundamental elements of functional analysis are presented. Although topological vector spaces are considered, essentially for the use of weak topologies, the focus is on normed spaces. Normed spaces in duality or metric duality form a convenient framework. The main pillars of functional analysis are presented: separatio...

We relate the second-order generalized derivatives of Chaney’s type to the classical second-order lower epiderivatives. The result sheds a new light on optimality conditions.

We study the Moreau regularization process for functions satisfying a general growth condition on general Banach spaces. We give differentiability criteria and we study the relationships between the subdifferentials of the function and the subdifferentials of its approximations. We also consider the Lasry-Lions process.

We study the Moreau regularization process for functions satisfying a general growth condition on general Banach spaces. We give di¤erentiability criteria and we study the relationships between the subdi¤erentials of the function and the subdi¤erentials of its approximations. We also consider the Lasry-Lions process. Mathematics Subject Classi…cati...

This textbook covers the main results and methods of real analysis in a single volume. Taking a progressive approach to equations and transformations, this book starts with the very foundations of real analysis (set theory, order, convergence, and measure theory) before presenting powerful results that can be applied to concrete problems. In additi...

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....

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 La...

We revisit a remarkable duality devoted to lower semicontinuous functions. We compare its definition in terms of a coupling with its definition in terms of linear-like (or elementary) functions. We consider several variants. Then, we deal with the passage from smoothness to rotundity and the reverse passage and we examine the transfer of boundednes...

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.

As a new illustration of the versatility of abstract subdifferentials we examine their introduction in the field of analysis of second-order generalized derivatives. We also consider some calculus rules for some of the various notions of such derivatives and we give an account of the effect of the Moreau regularization process.

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.

Second-order derivatives of Chaney’s type are introduced for an arbitrary subdifferential. Their uses for necessary optimality conditions and sufficient optimality conditions are put in light when the subdifferential satisfies a weak sum rule.

We consider some properties of the demand correspondence in the consumer theory such as nonemptiness of values, singlevaluedness, closedness. These results rely on a study of the passage from the utility function to the inverse utility function and on new notions of generalized concavity. They have obvious economic interpretations. Closedness for i...

As a new illustration of the versatility of abstract subdifferentials we examine their introduction in the field of analysis of second-order generalized derivatives. We also consider some calculus rules for some of the various notions of such derivatives and we give an account of the effect of the Moreau regularization process.

We look for an interpretation of the demand correspondence in the consumer theory as a generalized derivative of the inverse utility function. We test the main concepts of nonsmooth analysis for such an objective. The proofs only use classical methods in optimization such as penalization and optimality conditions.

Differential calculus is at the core of several sciences and techniques. Our world would not be the same without it: astronomy, electromagnetism, mechanics, optimization, thermodynamics, among others, use it as a fundamental tool.

We devote this opening chapter to some preliminary material dealing with sets, set-valued maps, convergences, estimates, and well-posedness.

The book present a comprehensive view of the main concepts dealing with non differentiable functions and non smooth sets. It contains preliminaries of independent interest making the subject accessible to non expert readers. These preliminary chapters can be used for separate courses.

Preface.- 1 Metric and Topological Tools.- 2 Elements of Differential Calculus.- 3 Elements of Convex Analysis.- 4 Elementary and Viscosity Subdifferentials.- 5 Circa-Subdifferentials, Clarke Subdifferentials.- 6 Limiting Subdifferentials.- 7 Graded Subdifferentials, Ioffe Subdifferentials.- References.- Index.

Given a closed subset S of a Banach space X, we study the minimal time function to reach a point x of X starting from S. We relate this problem to the study of the geodesic distance from x to S associated with a Riemannian metric or a Finsler metric. It appears that both cases can be treated simultaneously and yield solutions to a Hamilton-Jacobi e...

We give some new attention to the foundations of nonsmooth analysis. We endeavour to delineate the common features of usual subdifferentials. In particular, we stress calculus rules and properties linked with order. Our objective is to give the possibility of using subdifferentials without dealing with specific constructions.

We devote the present chapter to some fundamental notions of nonsmooth analysis upon which some other constructions can be built. Their main features are easy consequences of the definitions. Normal cones have already been considered in connection with optimality conditions. Here we present their links with subdifferentials for nonconvex, nonsmooth...

The fuzzy character of the rules devised in Chap. 4 incites us to pass to the limit. Such a process is simple enough in finite-dimensional Banach spaces. However, since a number of problems are set in functional spaces, one is led to examine what can be done in infinite-dimensional spaces. It appears that the situation may depend on the nature of t...

We devote the present chapter to one of the most famous attempts to generalize the concept of derivative. When limited to the class of locally Lipschitzian functions, it is of simple use, a fact that explains its success. The general case requires a more sophisticated approach. We choose a geometrical route to it involving the concept of normal con...

The class of convex functions is an important class that enjoys striking and useful properties. A homogenization procedure makes it possible to reduce this class to the subclass of sublinear functions. This subclass is next to the family of linear functions in terms of simplicity: the epigraph of a sublinear function is a convex cone, a notion almo...

In this last chapter we present an approach valid in every Banach space. The key idea, due to A.D. Ioffe, that yields such a universal theory consists in reducing the study to a convenient class of linear subspaces. Initially, Ioffe used the class of finite-dimensional subspaces of X [512, 513, 515, 516]; then he turned to the class of closed separ...

We study some classes of generalized affine functions, using a generalized differential. We study some properties and characterizations of these classes and we devise some characterizations of solution sets of optimization problems involving such functions or functions of related classes.

In this paper, some exact calculus rules are obtained for calculating the coderivatives of the composition of two multivalued maps. Similar rules are displayed for sums. A crucial role is played by an intermediate set-valued map called the resolvent. We first establish inclusions for contingent, Fréchet and limiting coderivatives. Combining them, w...

We provide a criterion giving a formula for the directional (or contingent) subdifferential of the difference of two convex functions. We even extend it to the difference of two approximately starshaped functions. Our analysis relies on a notion of approximate monotonicity for operators which is much less demanding than the usual one.

A notion of boundedly ε-lower subdifferentiable functions is introduced and investigated. It is shown that a bounded from below, continuous, quasiconvex function is locally boundedly ε-lower subdifferentiable for every ε>0. Some algorithms of cutting plane type are constructed to solve minimization problems with approximately lower subdifferentiabl...

We relate the image space approach and the theory of perturbations in the framework of nonconvex duality. We show that both theories can be applied to problems involving integral functionals on L1. Using simple characterizations of the Frechet subdifferential and of the limiting subdifferential of an integral functional on some L1 space, we show th...

We study some classes of generalized convex functions, using a generalized differential approach. By this we mean a set-valued mapping which stands either for a derivative, a subdifferential or a pseudo-differential in the sense of Jeyakumar and Luc. We establish some links between the corresponding classes of pseudoconvex, quasiconvex and another...

We present a simple proof of the separable reduction theorem, a crucial result of nonsmooth analysis which allows to extend to Asplund spaces the results known for separable spaces dealing with Fréchet subdifferentials. It relies on elementary results in convex analysis and avoids certain technicalities.

We raise some questions about duality theories in global optimization. The main one concerns the possibility to extend the use of conjugacies to general dualities for studying dual optimization problems. In fact, we examine whether dualities are the most general concepts to get duality results. We also consider the passage from a Lagrangian approac...

We associate a dual problem to a constrained optimization problem in which the objective is quasiconvex and either attains at 0 its global minimum or its global maximum. The attractive features of such a duality are that it does not require an additional parameter to set the dual and that the dual problem has a form which is similar to the one of t...

We present criteria for linear and nonlinear error bounds for lower semicontinuous functions on Banach spaces. We apply these criteria to various metric estimates which allow to give calculus rules for normal cones and subdifferentials.

The Ekeland duality scheme is a simple device. We examine its relationships with several classical dualities, such as the Fenchel–Rockafellar duality, the Toland duality, the Wolfe duality, and the quadratic duality. In particular, we show that the Clarke duality is a special case of the Ekeland duality scheme. Key wordsClarke duality-duality-Ekel...

Constraint qualifications are revisited, once again. These conditions are shown to be reminiscent of transversality theory. They are used as a useful tool for computing tangent cones, by the means of generalized inverse function theorems. The finite dimensional case is given a special treatment as the results are nicer and simpler in this case. Som...

In this paper, by virtue of two intermediate derivative-like multifunctions, which depend on an element in the intermediate space, some exact calculus rules are obtained for calculating the derivatives of the composition of two set-valued maps. Similar rules are displayed for sums. Moreover, by using these calculus rules, the solution map of a para...

We use representations of maximal monotone operators for studying recession (or asymptotic) operators associated to maximal monotone operators. Such a concept is useful for dealing with unboundedness.

We look for a simple general framework which would encompass the notion of symmetric self-dual spaces introduced by St. Simons [“From Hahn–Banach to monotonicity” (Lecture Notes in Mathematics 1693; Berlin: Springer) ( 2 2008; Zbl 1131.47050)] and the notion of self-paired product space proposed recently by the author [in: Lecture Notes in Economic...

We set up a formula for the Fréchet and ε-Fréchet subdifferentials of the difference of two convex functions. We even extend it to the difference of two approximately starshaped functions. As a consequence of this formula, we give necessary and sufficient conditions for local optimality in nonconvex optimization. Our analysis relies on the notion o...

We introduce a notion of continuity for multimaps (or set-valued maps) which is mild. It encompasses both lower semicontinuity and upper semicontinuity. We give characteri- zations and we consider some permanence properties. This notion can be used for various purposes. In particular, it is used for continuity properties of subdifferentials and of v...

Given a convergent sequence of Hamiltonians (Hn ) and a convergent sequence of initial data (gn ), we look for conditions ensuring that the sequences (un ) and (vn ) of Lax solutions and Hopf solutions respectively converge. The convergences we deal with are variational convergences. We take advantage of several recent results giving criteria for t...

A whole spectrum of subdifferentiability properties is delineated in which various degrees of uniformity are present. Related properties are introduced for sets. Some characterizations in terms of monotonicity properties are displayed.

We introduce new methods for defining generalized sums of monotone operators and generalized compositions of monotone operators with linear maps. Under asymptotic conditions we show these operations coincide with the usual ones. When the monotone operators are subdifferentials of convex functions, a similar conclusion holds. We compare these generali...

We study continuity properties of tangent and normal cones to a closed subset of a Banach space. Such a study can be seen as a nonsmooth set theoretic analogue of the study of continuously differentiable functions. We give particular attention to the case where different concepts coincide, a desirable feature such properties are able to offer. We i...

We study some classes of generalized convex functions, using a generalized derivative approach. We establish some links between these classes and we devise some optimality conditions for constrained optimization problems. In particular, we get Lagrange-Kuhn-Tucker multipliers for mathematical programming problems.

We look for a general framework in which the Ekeland duality can be formulated. We propose a scheme in which the parameter sets are provided with a coupling function which induces a conjugacy. The decision spaces are not supposed to have any special structure. We examine several examples. In particular, we consider some special classes of generaliz...

We prove that an approximately convex function on an open subset of an Asplund space is generically Fréchet differentiable, as are genuine convex functions. Thus, we give a positive answer to a question raised by S. Rolewicz. We also prove a more general result of that type for regular functions on an open subset of an Asplund space.

We study a class of functions which contains both convex functions and differentiable functions whose derivatives are locally Lipschitzian or Hölderian. This class is a subclass of the class of approximately convex functions. It enjoys refined properties. We also introduce a class of sets whose associated distance functions are of that type. We dis...

We study the possibility of defining tangent vectors to a metric space at a given point and tangent maps to applications from a metric space into another metric space. Such infinitesimal concepts may help in analysing situations in which no obvious differentiable structure is at hand. Some examples are presented; our interest arises from hyperspace...

We review various sorts of generalized convexity and we raise some questions about them. We stress the importance of some special subclasses of quasiconvex functions.

We present a survey of recent results about explicit solutions of the first-order Hamilton–Jacobi equation. We take advantage of the methods of asymptotic analysis, convex analysis and of nonsmooth analysis to shed a new light on classical results. We use formulas of the Hopf and Lax–Oleinik types. In the quasiconvex case the usual Fenchel conjugac...

In this work, we study some subdifferentials of the distance function to a nonempty nonconvex closed subset of a general Banach space. We relate them to the normal cone of the enlargements of the set which can be considered as regularizations of the set.

We present characterizations of some generalized convexity properties of functions with the help of a general subdifferential. We stress the case of lower semicontinuous functions. We also study the important case of marginal functions and we provide representation results.

We study functions whose directional derivatives can be written as a difference of two extended real valued sublinear function (ds functions). The pair of convex sets obtained as the support functions of these sublinear functions is provided by a familiar equivalence relation and is considered as the bi-subdifferential of the function. We study cal...

The relationships between the semismoothness of a function and the submonotonicity of its subdieren- tials at some given point are studied. A notion of approximate starshapedness at that point is introduced and compared with these properties. Some criteria ensuring that dierent subdierentials

We study two classes of sets whose associated distance functions satisfy properties akin to approx-imate convexity. Since the class of approximately convex functions is known to enjoy nice properties, one may expect analogous properties for this class of sets. We present characterizations and delineate links with the concept of approximately convex...

We study the convergence of maximal monotone operators with the help of representations by convex functions. In particular, we prove the convergence of a sequence of sums of maximal monotone operators under a general qualification condition of the Attouch-Brezis type.

We study an approximation method for sets and functions which erases comers but keeps smooth parts. Basic properties of such a method are pointed out in a general and simple way. Several convergence results are provided, essentially in the framework of variational analysis.

The existence of Lagrange-Karush-Kuhn-Tucker multipliers in differentiable mathematical programming is shown to be a direct consequence of fundamental rules for computing tangent cones. The relationships of these rules with the transversality conditions of differential topology are pointed out. These rules also bear some connections with subdiffere...

We present necessary and sufficient optimality conditions for a problem with a convex set constraint and a quasiconvex objective function. We apply the obtained results to a mathematical programming problem involving quasiconvex functions.

Two classes of functions encompassing the cone of convex functions and the space of strictly differentiable functions are presented and compared. Related properties for sets and multimappings are dealt with.

There is a recent surge of interest for the representation of monotone operators by convex functions. It can explained by the success of convex analysis in obtaining the fundamental results about maximal monotone operators. Convex analysis can also be combined with variational analysis to get new convergence results. Here we take another direction...

We study the convergence of maximal monotone operators with the help of repre- sentations by convex functions. In particular, we prove the convergence of the sums of sequences of maximal monotone operators under a general qualification condition of the Attouch-Brezis type.

A weakened notion of multivalued contraction mapping is introduced. Some fixed point results relying on this notion are presented. The associated fixed points sets are shown to enjoy a Lipschitzian behaviour with respect to the graphs of the multifunctions. Application are given to the dependence of solutions of differential inclusions of the form...

We provide conditions ensuring the existence of a solution to a noncoercive minimization problem. These conditions are also necessary. They enlarge the applicability of previous results due to Baiocchi, Buttazzo, Gastaldi and Tomarelli and their extensions by Auslender. A particular attention is given to the case of quasiconvex objective functions.

We give some variants of a constant associated to a real-valued function on a Banach space and a point of the space as introduced by S. Fitzpatrick. They have a one-sided character and are related to different types of differentiability or subdifferentiability.

We introduce one-sided versions of Lagrangians and perturbations. We relate them, using concepts from generalized convexity. In such a way, we are able to present the main features of duality theory in a general framework encompassing numerous special instances. We focus our attention on the set of multipliers. We look for an interpretation of mult...

We combine a Lagrangian approach inspired by convex and quasiconvex dualities with a penalization approach to mathematical programming. We use the ideas of abstract convexity. We focus our attention on the set of multipliers. We look for an interpretation of multipliers as elements of generalized subdifferentials of the performance function associa...

We answer a few questions raised by S. Fitzpatrick concerning the representation of maximal monotone operators by convex functions. We also examine some other questions concerning this representation and other ones which have recently emerged.

We answer a few questions raised by S. Fitzpatrick concerning the representation of maximal monotone operators by convex functions. We also examine some other questions concerning this representation and other ones which have recently emerged.

We compare various notions of approximations of sets. Several of them are one-sided versions of existing notions. We devote a particular attention to the case where the approximating set is a translated cone. We point out some consequences for nonsmooth analysis and optimization.

We consider some elements of the influence of methods from convex analysis and duality on the study of Hamilton-Jacobi equations.

We present a survey of some uses of a remarkable convergence on families of sets or functions. We evoke some of its applications and stress some calculus rules. The main novelty lies in the use of a notion of “firm” (or uniform) asymptotic cone to an unbounded subset of a normed space. This notion yields criteria for the study of boundedness proper...

We present some observations about links between some classical theories of microeconomics and dualities which have been used in optimization theory and in the study of first-order Hamilton-Jacobi equations. We introduce a variant of the classical indirect utility function called the wary indirect utility function and a variant of the expenditure f...

We introduce a notion of conjugacy for relations between a normed vector space and the real field. When the correspondence is a Legendre function, one recovers the Legendre transform.

We present a new inverse mapping theorem for correspondences. It uses a notion of differentiability for multifunctions which seems to be new. We compare it with previous versions. We provide an application to differential inclusions.

We prove that the projection operator on a nonempty closed convex subset C of a uniformly convex Banach spaces is uniformly continuous on bounded sets and we provide an estimate of its modulus of uniform continuity. We derive this result from a study of the dependence of the projection on C of a given point when C varies.