Cyclic Hypomonotonicity,Cyclic Submonotonicity,and Integration

Rockafellar has shown that the subdifferentials of convex functions are always cyclically monotone operators. Moreover, maximal cyclically monotone operators are necessarily operators of this type, since one can construct explicitly a convex function, which turns out to be unique up to a constant, whose subdifferential gives back the operator. This result is a cornerstone in convex analysis and relates tightly convexity and monotonicity. In this paper, we establish analogous robust results that relate weak convexity notions to corresponding notions of weak monotonicity, provided one deals with locally Lipschitz functions and locally bounded operators. In particular, the subdifferentials of locally Lipschitz functions that are directionally hypomonotone [respectively, directionally submonotone] enjoy also an additional cyclic strengthening of this notion and in fact are maximal under this new property. Moreover, every maximal cyclically hypomonotone [respectively, maximal cyclically submonotone] operator is always the Clarke subdifferential of some directionally weakly convex [respectively, directionally approximately convex] locally Lipschitz function, unique up to a constant, which in finite dimentions is a lower C² function [respectively, a lower C¹ function].     

Subdifferential characterization of approximate convexity: the lower semicontinuous case

It is known that a locally Lipschitz function is approximately convex if, and only if, its Clarke subdifferential is a submonotone operator. The main object of this work is to extend the above characterization to the class of lower semicontinuous functions. To this end, we establish a new approximate mean value inequality involving three points. We also show that an analogue of the Rockafellar maximal monotonicity theorem holds for this class of functions and we discuss the case of arbitrary subdifferentials.     

Generalized gradients of monotone type

We examine the properties of the subdifferential in the sense of Clarke of certain locally Lipschitz, quasi-convex functions. We prove that, even if they may not possess a pseudomonotone-type subdifferential, if we consider the operator A+∂f, where A is an operator of type (S)₊, then the sum is pseudomonotone. A new type of subdifferential for Lipschitz functions is also presented. We prove some calculus rules and we establish that in the context of reflexive Banach spaces is an operator of type (M).     

Approximation and decomposition properties of some classes of locally D.C. functions

We study the connexion between local and global decompositions of some important subclasses of locally d.c. functions (functions which locally split as a difference of two convex functions). Then we tackle the problem of regularizing such functions by the Moreau-Yosida process and prove in particular that the class of lower-C ² functions fits well this approximation procedure.     

Integration of multivalued operators and cyclic submonotonicity

We introduce a notion of cyclic submonotonicity for multivalued operators from a Banach space to its dual. We show that if the Clarke subdifferential of a locally Lipschitz function is strictly submonotone on an open subset of , then it is also maximal cyclically submonotone on , and, conversely, that every maximal cyclically submonotone operator on is the Clarke subdifferential of a locally Lipschitz function, which is unique up to a constant if is connected. In finite dimensions these functions are exactly the lower C functions considered by Spingarn and Rockafellar.

     

Local Lipschitz-constant Functions and Maximal Subdifferentials

It is shown that if k(x) is an upper semicontinuous and quasi lower semicontinuous function on a Banach space X, then k(x)B _X* is the Clarke subdifferential of some locally Lipschitz function on X. Related results for approximate subdifferentials are also given. Moreover, on smooth Banach spaces, for every locally Lipschitz function with minimal Clarke subdifferential, one can obtain a maximal Clarke subdifferential map via its local Lipschitz-constant function. Finally, some results concerning the characterization and calculus of local Lipschitz-constant functions are developed.     

Asplund sets, differentiability and subdifferentiability of functions in Banach spaces

We show that Asplund sets are effective tools to study differentiability of Lipschitz functions, and ε-subdifferentiability of lower semicontinuous functions on general Banach spaces. If a locally Lipschitz function defined on an Asplund generated space has a minimal Clarke subdifferential mapping, then it is TBY-uniformly strictly differentiable on a dense Gδ subset of X. Examples are given of locally Lipschitz functions that are TBY-uniformly strictly differentiable everywhere, but nowhere Fréchet differentiable.     

Relation between Demyanov difference and Minkowski difference of convex compact subsets inR 2

A necessary and sufficient condition for Demyanov difference and Minkowski difference of compact convex subsets inR ² being equal is given in this paper. Several examples are computed by Matlab to test our result. The necessary and sufficient condition makes us to compute Clarke subdifferential by quasidifferential for a special of Lipschitz functions.     

Morse theory for some lower-C 2 functions in finite dimension

Using inf-regularization methods, we prove that Morse inequalities hold for some lower-C ² functions. For this purpose, we first recall some properties of the class of lower-C ² functions and of their Moreau-Yosida approximations. Then, we establish, under some qualification conditions on the critical points, that it is possible to define a Morse index for a lower-C ² functionf. This index is preserved by the Moreau-Yosida approximation process. We prove in particular that the Moreau-Yosida approximations are twice continuolusly differentiable around such a critical point which is shown to be a strict local minimum of the restriction off and of its approximations to some affine space. In a last step, Morse inequalities are written for Moreau-Yosida approximations and with the aid of deformation retractions we prove that these inequalities also hold for some lower-C ² functions.     

“Two Nontrivial Critical Points for Nonsmooth Functionals via Local Linking and Applications”

In this paper, we extend to nonsmooth locally Lipschitz functionals the multiplicity result of Brezis–Nirenberg (Communication Pure Applied Mathematics and 44 (1991)) based on a local linking condition. Our approach is based on the nonsmooth critical point theory for locally Lipschitz functions which uses the Clarke subdifferential. We present two applications. This first concerns periodic systems driven by the ordinary vector p-Laplacian. The second concerns elliptic equations at resonance driven by the partial p-Laplacian with Dirichlet boundary condition. In both cases the potential function is nonsmooth, locally Lipschitz.     

The Lipschitzianity of convex vector and set-valued functions

It is well known that every scalar convex function is locally Lipschitz on the interior of its domain in finite dimensional spaces. The aim of this paper is to extend this result for both vector functions and set-valued mappings acting between infinite dimensional spaces with an order generated by a proper convex cone C. Under the additional assumption that the ordering cone C is normal, we prove that a locally C-bounded C-convex vector function is Lipschitz on the interior of its domain by two different ways. Moreover, we derive necessary conditions for Pareto minimal points of vector-valued optimization problems where the objective function is C-convex and C-bounded. Corresponding results are derived for set-valued optimization problems.     

A Remark on the Local Lipschitz Continuity of Vector Hysteresis Operators

It is known that the vector stop operator with a convex closed characteristic Z of class C ¹ is locally Lipschitz continuous in the space of absolutely continuous functions if the unit outward normal mapping n is Lipschitz continuous on the boundary Z of Z. We prove that in the regular case, this condition is also necessary.     

Lipschitz B-Vex Functions and Nonsmooth Programming

In this paper, the equivalence between the class of B-vex functions and that of quasiconvex functions is proved. Necessary and sufficient conditions, under which a locally Lipschitz function is B-vex, are established in terms of the Clarke subdifferential. Regularity of locally Lipschitz B-vex functions is discussed. Furthermore, under appropriate conditions, a necessary optimality condition of the Slater type and a sufficient optimality condition are obtained for a nonsmooth programming problem involving B-vex functions.     

Semilinear hemivariational inequalities at resonance

In this paper we examine a semilinear hemivariational inequality at resonance in the first eigenvalue λ₁ of (−Δ,H ₀ ¹ (Z)). We prove two existence theorems for such problems. Our approach is variational and is based on the nonsmooth critical point theory of Chang, which uses the subdifferential calculus of Clarke for locally Lipschitz functions.     

A differential operator and weak topology for Lipschitz maps

We show that the Scott topology induces a topology for real-valued Lipschitz maps on Banach spaces which we call the L-topology. It is the weakest topology with respect to which the L-derivative operator, as a second order functional which maps the space of Lipschitz functions into the function space of non-empty weak^∗ compact and convex valued maps equipped with the Scott topology, is continuous. For finite dimensional Euclidean spaces, where the L-derivative and the Clarke gradient coincide, we provide a simple characterization of the basic open subsets of the L-topology. We use this to verify that the L-topology is strictly coarser than the well-known Lipschitz norm topology. A complete metric on Lipschitz maps is constructed that is induced by the Hausdorff distance, providing a topology that is strictly finer than the L-topology but strictly coarser than the Lipschitz norm topology. We then develop a fundamental theorem of calculus of second order in finite dimensions showing that the continuous integral operator from the continuous Scott domain of non-empty convex and compact valued functions to the continuous Scott domain of ties is inverse to the continuous operator induced by the L-derivative. We finally show that in dimension one the L-derivative operator is a computable functional.     

Extremality in Solving General Quasilinear Parabolic Inclusions

The paper deals with an initial boundary-value problem for a parabolic inclusion whose multivalued term has the structure of a difference between the Clarke generalized gradient of some locally Lipschitz function verifying a unilateral growth condition and the subdifferential of a convex function, and where the elliptic part is expressed by a general quasilinear operator of the Leray-Lions type. Our results address not only the existence of solutions, but also the extremality inside an order interval determined by appropriately defined upper and lower solutions as well as the compactness of the solution set in suitable spaces.     

Characterization and recognition of d.c. functions

A function ${f : \Omega \to \mathbb{R}}$ , where Ω is a convex subset of the linear space X, is said to be d.c. (difference of convex) if f =  g ? h with ${g, h : \Omega \to \mathbb{R}}$ convex functions. While d.c. functions find various applications, especially in optimization, the problem to characterize them is not trivial. There exist a few known characterizations involving cyclically monotone set-valued functions. However, since it is not an easy task to check that a given set-valued function is cyclically monotone, simpler characterizations are desired. The guideline characterization in this paper is relatively simple (Theorem 2.1), but useful in various applications. For example, we use it to prove that piecewise affine functions in an arbitrary linear space are d.c. Additionally, we give new proofs to the known results that C ^1,1 functions and lower-C ² functions are d.c. The main goal remains to generalize to higher dimensions a known characterization of d.c. functions in one dimension: A function ${f : \Omega \to \mathbb{R}, \Omega \subset \mathbb{R}}$ open interval, is d.c. if and only if on each compact interval in Ω the function f is absolutely continuous and has a derivative of bounded variation. We obtain a new necessary condition in this direction (Theorem 3.8). We prove an analogous sufficient condition under stronger hypotheses (Theorem 3.11). The proof is based again on the guideline characterization. Finally, we obtain results concerning the characterization of convex and d.c. functions obeying some kind of symmetry.     

New Formulas for the Fenchel Subdifferential of the Conjugate Function

Following (López and Volle, J Convex Anal 17, 2010) we provide new formulas for the Fenchel subdifferential of the conjugate of functions defined on locally convex spaces. In particular, this allows deriving expressions for the minimizers set of the lower semicontinuous convex hull of such functions. These formulas are written by means of primal objects related to the subdifferential of the initial function, namely a new enlargement of the Fenchel subdifferential operator.     

Moreau–Yosida regularization of Lagrangian-dual functions for a class of convex optimization problems

In this paper, we consider the Lagrangian dual problem of a class of convex optimization problems, which originates from multi-stage stochastic convex nonlinear programs. We study the Moreau–Yosida regularization of the Lagrangian-dual function and prove that the regularized function η is piecewise C ², in addition to the known smoothness property. This property is then used to investigate the semismoothness of the gradient mapping of the regularized function. Finally, we show that the Clarke generalized Jacobian of the gradient mapping is BD-regular under some conditions.      

Subdifferentials of compactly lipschitzian vector-valued functions

Summary We introduce the concept of compactly lipschitzian functions taking values in a topological vector space F. We show that if F is finite dimensional the Lipschitz functions are compactly lipschitizian. We define the notions of generalized directional derivatives and subdifferentials for such functionsf taking values in an ordered topological vector space. It is shown that this notion of subdifferential coincides with the one of F. H. Clarke whenf is Lispchits and F=. Some formulas for this subdifferential concerning the cases of finite sum, composition, pointwise supremum and continuous sum are studied.