Approximate convexity and submonotonicity in locally convex spaces (original) (raw)
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Approximate convexity and submonotonicity
Journal of Mathematical Analysis and Applications, 2004
It is shown that a locally Lipschitz function is approximately convex if, and only if, its Clarke subdifferential is a submonotone operator. Consequently, in finite dimensions, the class of locally Lipschitz approximately convex functions coincides with the class of lower-C 1 functions. Directional approximate convexity is introduced and shown to be a natural extension of the class of lower-C 1 functions in infinite dimensions. The following characterization is established: a multivalued operator is maximal cyclically submonotone if, and only if, it coincides with the Clarke subdifferential of a locally Lipschitz directionally approximately convex function, which is unique up to a constant. Furthermore, it is shown that in Asplund spaces, every regular function is generically approximately convex.
Characterization of the subdifferentials of convex functions
Pacific Journal of Mathematics, 1966
Each lower semi-continuous proper convex function / on a Banach space E defines a certain multivalued mapping df from E to E* called the subdifferential of /. It is shown here that the mappings arising this way are precisely the ones whose graphs are maximal "cyclically monotone" relations on E X E*, and that each of these is also a maximal monotone relation. Furthermore, it is proved that df determines / uniquely up to an additive constant. These facts generally fail to hold when E is not a Banach space. The proofs depend on establishing a new result which relates the directional derivatives of / to the existence of approximate subgradients.
Convex functions, subdifferentiability and renormings
Acta Mathematica Sinica, 1998
This paper through discussing subdifferentiability and convexity of convex functions shows that a Banach space admits an equivalent uniformly [locally uniformly, strictly] convex norm if and only if there exists a continuous uniformly [locally uniformly, strictly] convex function on some nonempty open convex subset of the space and presents some characterizations of super-reflexive Banach spaces.
Lipschitz Continuity of Convex Functions
Applied Mathematics & Optimization, 2020
We provide some necessary and sufficient conditions for a proper lower semicontinuous convex function, defined on a real Banach space, to be locally or globally Lipschitz continuous. Our criteria rely on the existence of a bounded selection of the subdifferential mapping and the intersections of the subdifferential mapping and the normal cone operator to the domain of the given function. Moreover, we also point out that the Lipschitz continuity of the given function on an open and bounded (not necessarily convex) set can be characterized via the existence of a bounded selection of the subdifferential mapping on the boundary of the given set and as a consequence it is equivalent to the local Lipschitz continuity at every point on the boundary of that set. Our results are applied to extend a Lipschitz and convex function to the whole space and to study the Lipschitz continuity of its Moreau envelope functions.
Subdifferential Characterization of Quasiconvexity and Convexity
Journal of Convex Analysis, 1994
Let f: X! IR f+1g be a lower semicontinuous function on a Banach space X. We show that f is quasiconvex if and only if its Clarke subdi erential@ f is quasimonotone. As an immediate consequence, we get that f is convex if and only if@ f is monotone.
Approximately convex functions and approximately monotonic operators
Nonlinear Analysis-theory Methods & Applications, 2007
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.
Optimization
Given a set H of functions defined on a set X,à function f : X Þ Ñ R is called abstract H-convex if it is the upper envelope of its H-minorants, i.e., such its minorants which belong to the set H; and f is called regularly abstract H-convex if it is the upper envelope of its maximal (with respect to the pointwise ordering) H-minorants. In the paper we first present the basic notions of (regular) H-convexity for the case when H is an abstract set of functions. For this abstract case a general sufficient condition based on Zorn's lemma for a H-convex function to be regularly H-convex is formulated. The goal of the paper is to study the particular class of regularly H-convex functions, when H is the set L p CpX, Rq of real-valued Lipschitz continuous classically concave functions defined on a real normed space X. For an extended-real-valued function f : X Þ Ñ R to be L p C-convex it is necessary and sufficient that f be lower semicontinuous and bounded from below by a Lipschitz continuous function; moreover, each L p C-convex function is regularly L p C-convex as well. We focus on L p Csubdifferentiability of functions at a given point. We prove that the set of points at which an L p C-convex function is L p C-subdifferentiable is dense in its effective domain. This result extends the well-known classical Brøndsted-Rockafellar theorem on the existence of the subdifferential for convex lower semicontinuous functions to the more wide class of lower semicontinuous functions. Using the subset L p C θ of the set L p C consisting of such Lipschitz continuous concave functions that vanish at the origin we introduce the notions of L p C θ-subgradient and L p C θ-subdifferential of a function at a point which generalize the corresponding notions of the classical convex analysis. Some properties and simple calculus rules for L p C θ-subdifferentials as well as L p C θ-subdifferential conditions for global extremum points are established. Symmetric notions of abstract L q C-concavity and L q C-superdifferentiability of functions where L q C :" L q CpX, Rq is the set of Lipschitz continuous convex functions are also considered.