The approximate subdifferential of composite functions

This paper deals with the approximate subdifferential chain rule in a Banach space. It establishes specific results when the real-valued function is locally Lipschitzian and the mapping is strongly compactly Lipschitzian.

Regularity and strong sufficient optimality conditions in differentiable optimization problems

This paper studies the metric regularity of multivalued functions on Banach spaces, tangential approximations of the feasible set and strong sufficient optimality conditions of a parametrized optimization problem minimize The results are applied to the tangent approximations and the local stability properties of solutions of this perturbed optimization problem.

Tangency conditions for multivalued mappings

We prove that interiority conditions imply tangency conditions for two multivalued mappings from a topological space into a normed vector space. As a consequence, we obtain the lower semicontinuity of the intersection of two multivalued mappings. An application to the epi-upper semicontinuity of the sum of convex vector-valued mappings is given.

Qualification Conditions for Calculus Rules of Coderivatives of Multivalued Mappings

AbstractThis paper establishes by a general approach a full calculus for the limiting Fréchet and the approximate coderivatives of multivalued mappings. This approach allows us to produce several new verifiable qualification conditions for such calculus rules.

Noncoincidence of Approximate and Limiting Subdifferentials of Integral Functionals

For a locally Lipschitz integral functional $I_f$ on $L^1(T,\mathbf{R}^n)$ associated with a measurable integrand f, the limiting subdifferential and the approximate subdifferential never coincide at a point $x_0$ where $f(t,\cdot)$ is not subdifferentially regular at $x_0(t)$ for a.e. $t\in T$. The coincidence of both subdifferentials occurs on a dense set of $L^1(T,\mathbf{R}^n)$ if and only if $f(t,\cdot)$ is convex for a.e. $t\in T$. Our results allow us to characterize Aubin's Lipschitz-like property as well as the convexity of multivalued mappings between $L^1$-spaces. New necessary optimality conditions for some Bolza problems are also obtained.

Characterization of the Clarke regularity of subanalytic sets

International audience; In this note, we will show that for a closed subanalytic subset $A \subset \mathbb{R}^n$, the Clarke tangential regularity of $A$ at $x_0 \in A$ is equivalent to the coincidence of the Clarke's tangent cone to $A$ at $x_0$ with the set \\$$\mathcal{L}(A, x_0):= \bigg\{\dot{c}_+(0) \in \mathbb{R}^n: \, c:[0,1]\longrightarrow A\;\;\mbox{\it is Lipschitz}, \, c(0)=x_0\bigg\}.$$Where $\dot{c}_+(0)$ denotes the right-strict derivative of $c$ at $0$. The results obtained are used to show that the Clarke regularity of the epigraph of a function may be characterized by a new formula of the Clarke subdifferential of that function.

On the Calmness of a Class of Multifunctions

The paper deals with the calmness of a class of multifunctions in finite dimensions. Its first part is devoted to various conditions for calmness, which are derived in terms of coderivatives and subdifferentials. The second part demonstrates the importance of calmness in several areas of nonsmooth analysis. In particular, we focus on nonsmooth calculus and solution stability in mathematical programming and in equilibrium problems. The derived conditions find a number of applications there.

Approximations and Metric Regularity in Mathematical Programming in Banach Space

This paper establishes verifiable conditions ensuring the important notion of metric regularity for general nondifferentiable programming problems in Banach spaces. These conditions are used to obtain Lagrange-Kuhn-Tucker multipliers for minimization problems with infinitely many inequality and equality constraints.

Necessary Optimality Conditions in Multiobjective Dynamic Optimization

We consider a nonsmooth multiobjective optimal control problem related to a general preference. Both differential inclusion and endpoint constraints are involved. Necessary conditions and Hamiltonian necessary conditions expressed in terms of the limiting Frechet subdifferential are developed. Examples of useful preferences are given.

Differential properties of the Moreau envelope

International audience; In a vector space endowed with a uniformly Gâteaux differentiable norm, it is proved that the Moreau envelope enjoys many remarkable differential properties and that its subdifferential can be completely described through a certain approximate proximal mapping. This description shows in particular that the Moreau envelope is essentially directionally smooth. New differential properties are derived for the distance function associated with a closed set. Moreover, the analysis, when applied to the investigation of the convexity of Tchebyshev sets, allows us to recover several known results in the literature and to provide some new ones.

Metric regularity and subdifferential calculus in Banach spaces

In this paper we give verifiable conditions in terms of limiting Frechet subdifferentials ensuring the metric regularity of a multivalued functionF(x)=−g(x)+D. We apply our results to the study of the limiting Frechet subdifferential of a composite function defined on a Banach space.

On a class of compactly epi-Lipschitzian sets

The paper is devoted to the study of the so-called compactly epi-Lipschitzian sets. These sets are needed for many aspects of generalized differentiation, particulary for necessary optimality conditions, stability of mathematical programming problems and calculus rules for subdifferentials and normal cones. We present general conditions under which sets defined by general constraints are compactly epi-Lipschitzian. This allows us to show how the compact epi-Lipschitzness properties behave under set intersections.

Controllability and strong controllability of differential inclusions

Abstract In this paper, we prove sufficient conditions for controllability and strong controllability in terms of the Mordukhovich subdifferential for two classes of differential inclusions. The first one is the class of sub-Lipschitz multivalued functions introduced by Loewen–Rockafellar (1994) [10] . The second one, introduced recently by Clarke (2005) [18] , is the class of multivalued functions which are pseudo-Lipschitz and satisfy the so-called tempered growth condition. To do this, we establish an error bound result in terms of the Mordukhovich subdifferential outside Asplund spaces.

Qualification conditions for multivalued functions in Banach spaces with applications to nonsmooth vector optimization problems

In this paper we introduce qualification conditions for multivalued functions in Banach spaces involving the A-approximate subdifferential, and we show that these conditions guarantee metric regularity of multivalued functions. The results are then applied for deriving Lagrange multipliers of Fritz—John type and Kuhn—Tucker type for infinite non-smooth vector optimization problems.

A general metric regularity in asplund banach spaces

This paper establishes a simple and easily-applied criterion for determining whether a multivalued mapping is metrically regular relatively to a subset in the range space.

Coderivatives of multivalued mappings, locally compact cones and metric regularity

Necessary conditions for extremality and separation theorems with applications to multiobjective optimization

The aim of this paper is to give necessary conditions for extremality in terms of an abstract subdifferential and to obtain general separation theorems including both finite and infinite classical separation theorems. This approach, which is mainly based on Ekeland's variational principle and the concept of locally weak-star compact cones, can be considered as a generalization f the notions of optima in problems of scalar or vector optimization with and without constraints. The results obtained are applied to derive new necessary optimality conditions for Pareto local minimum and weak Pareto minimum of nonsmooth multlobjectivep rogramming problems.

Weak regularity of functions and sets in Asplund spaces

Abstract In this paper, we study a new concept of weak regularity of functions and sets in Asplund spaces. We show that this notion includes prox-regular functions, functions whose subdifferential is weakly submonotone and amenable functions in infinite dimension. We establish also that weak regularity is equivalent to Mordukhovich regularity in finite dimension. Finally, we give characterizations of the weak regularity of epi-Lipschitzian sets in terms of their local representations.

Controlled polyhedral sweeping processes: existence, stability, and optimality conditions

This paper is mainly devoted to the study of controlled sweeping processes with polyhedral moving sets in Hilbert spaces. Based on a detailed analysis of truncated Hausdorff distances between moving polyhedra, we derive new existence and uniqueness theorems for sweeping trajectories corresponding to various classes of control functions acting in moving sets. Then we establish quantitative stability results, which provide efficient estimates on the sweeping trajectory dependence on controls and initial values. Our final topic, accomplished in finite-dimensional state spaces, is deriving new necessary optimality and suboptimality conditions for sweeping control systems with endpoint constrain…

C 1,ω (·) -regularity and Lipschitz-like properties of subdifferential

Envelopes for sets and functions II: generalized polarity and conjugacy

International audience; Let X,Y be two nonempty sets, Φ an extended real-valued bivariate coupling function on X × Y and Γ a subset of X × Y. The present paper provides extensions to the well-known generalized Φ-conjugacy and Γ-polarity of diverse results of our previous work [2] related to φ-conjucacy and Λ-polarity, where Λ is a subset of a vector space E and φ is a function on E defining the particular coupling function (x,y)→φ(x−y) on E × E. A particular attention is devoted to the conjugacy functions (resp. polarity sets) which are mutually generating. Finally, for a superadditive conjugacy function Φ, we obtain a full description of the class of Φ-envelopes.

Prices and Pareto optima

We provide necessary conditions for Pareto optimum in economies where tastes or technologies may be nonconvex, nonsmooth, and affected by externalities. Firms can pursue own objectives, much like the consumers. Infinite-dimensional commodity spaces are accommodated. Public goods and material balances are accounted for as special instances of linear restrictions.

Strategic behavior and partial cost sharing

Abstract The main objects here are games in which players mainly compete but nonetheless collaborate on some subsidiary activities. Play assumes a two-stage nature in that first-stage moves presume coordination of some subsequent tasks. Specifically, we consider instances where second-stage coordination amounts to partial cost sharing, anticipated and sustained as a core solution. Examples include regional Cournot oligopolies with joint transportation. We define and characterize equilibria, and inquire about their existence.

The validity of the “liminf” formula and a characterization of Asplund spaces

Abstract We show that for a given bornology β on a Banach space X the following “ lim inf ” formula lim inf x ′ ⟶ C x T β ( C ; x ′ ) ⊂ T c ( C ; x ) holds true for every closed set C ⊂ X and any x ∈ C , provided that the space X × X is ∂ β -trusted. Here T β ( C ; x ) and T c ( C ; x ) denote the β-tangent cone and the Clarke tangent cone to C at x. The trustworthiness includes spaces with an equivalent β-differentiable norm or more generally with a Lipschitz β-differentiable bump function. As a consequence, we show that for the Frechet bornology, this “ lim inf ” formula characterizes in fact the Asplund property of X. We use our results to obtain new characterizations of T β -pseudoconve…

A differential equation approach to implicit sweeping processes

International audience; In this paper, we study an implicit version of the sweeping process. Based on methods of convex analysis, we prove the equivalence of the implicit sweeping process with a differential equation, which enables us to show the existence and uniqueness of the solution to the implicit sweeping process in a very general framework. Moreover, this equivalence allows us to give a characterization of nonsmooth Lyapunov pairs and invariance for implicit sweeping processes. The results of the paper are illustrated with two applications to quasistatic evolution variational inequalities and electrical circuits.

A note on Fréchet and approximate subdifferentials of composite functions

The aim of this note is to present in the reflexive Banach space setting a natural and simple proof of the formula of the approximate subdifferential of a composite function.

Characterizations of convex approximate subdifferential calculus in Banach spaces

International audience; We establish subdifferential calculus rules for the sum of convex functions defined on normed spaces. This is achieved by means of a condition relying on the continuity behaviour of the inf-convolution of their corresponding conjugates, with respect to any given topology intermediate between the norm and the weak* topologies on the dual space. Such a condition turns out to also be necessary in Banach spaces. These results extend both the classical formulas by Hiriart-Urruty and Phelps and by Thibault.

Subdifferential and conjugate calculus of integral functions with and without qualification conditions

We characterize the subdifferential and the Fenchel conjugate of convex integral functions by means of respectively the approximate subdifferential and the conjugate of the associated convex normal integrands. The results are stated in Suslin locally convex spaces, and do not require continuity-type qualification conditions on the functions, nor special topological or algebraic structures on the index set. Consequently, when confined to separable Banach spaces, the characterizations of such a subdifferential are obtained using only the exact subdifferential of the given integrand but at nearby points. We also provide some simplifications of our formulas when additional continuity conditions…

Geometric characterizations of the strict Hadamard differentiability of sets

Let $S$ be a closed subset of a Banach space $X$. Assuming that $S$ is epi-Lipschitzian at $\bar{x}$ in the boundary $ \bd S$ of $S$, we show that $S$ is strictly Hadamard differentiable at $\bar{x}$ IFF the Clarke tangent cone $T(S, \bar{x})$ to $S$ at $\bar{x}$ contains a closed hyperplane IFF the Clarke tangent cone $T(\bd S, \bar{x})$ to $\bd S$ at $\bar{x}$ is a closed hyperplane. Moreover when $X$ is of finite dimension, $Y$ is a Banach space and $g: X \mapsto Y$ is a locally Lipschitz mapping around $\bar{x}$, we show that $g$ is strictly Hadamard differentiable at $\bar{x}$ IFF $T(\mathrm{graph}\,g, (\bar{x}, g(\bar{x})))$ is isomorphic to $X$ IFF the set-valued mapping $x\rightrigh…

Hoffman's Error Bound, Local Controllability, and Sensitivity Analysis

Our aim is to present sufficient conditions ensuring Hoffman's error bound for lower semicontinuous nonconvex inequality systems and to analyze its impact on the local controllability, implicit function theorem for (non-Lipschitz) multivalued mappings, generalized equations (variational inequalities), and sensitivity analysis and on other problems like Lipschitzian properties of polyhedral multivalued mappings as well as weak sharp minima or linear conditioning. We show how the information about our sufficient conditions can be used to provide a computable constant such that Hoffman's error bound holds. We also show that this error bound is nothing but the classical Farkas lemma for linear …

Metric regularity for strongly compactly Lipschitzian mappings

Chain rules for coderivatives of multivalued mappings in Banach spaces

Metric regularity and second-order necessary optimality conditions for minimization problems under inclusion constraints

In this paper, we establish some general metric regularity results for multivalued functions on Banach spaces. Then, we apply them to derive second-order necessary optimality conditions for the problem of minimizing a functionf on the solution set of an inclusion 0?F(x) withx?C, whenF has a closed convex second-order derivative.

Constraint qualifications and Lagrange multipliers in nondifferentiable programming problems

In this paper, we present several constraint qualifications, and we show that these conditions guarantee the nonvacuity and the boundedness of the Lagrange multiplier sets for general nondifferentiable programming problems. The relationships with various constraint qualifications are investigated.