The first part of the paper provides new characterizations of the normal cone to the effective domain of the supremum of an arbitrary family of convex functions. These results are applied in the second part to give new formulas for the subdifferential of the supremum function, which use both the active and nonactive functions at the reference point. Only the data functions are involved in these characterizations, the active ones from one side, together with the nonactive functions multiplied by some appropriate parameters. In contrast with previous works in the literature, the main feature of our subdifferential characterization is that the normal cone to the effective domain of the supremum (or to finite-dimensional sections of this domain) does not appear. A new type of optimality conditions for convex optimization is established at the end of the paper.
This work provides calculus for the Fréchet and limiting subdifferential of the pointwise supremum given by an arbitrary family of lower semicontinuous functions. We start our study showing fuzzy results about the Fréchet subdifferential of the supremum function. Posteriorly, we study in finite- and infinite-dimensional settings the limiting subdifferential of the supremum function. Finally, we apply our results to the study of the convex subdifferential; here we recover general formulae for the subdifferential of an arbitrary family of convex functions.
The aim of this work is to provide formulae for the subdifferential and the conjungate function of the supremun function over an arbitrary family of functions. The work is principally motivated by the case when data functions are lower semicontinuous proper and convex. Nevertheless, we explore the case when the family of functions is arbitrary, but satisfying that the biconjugate of the supremum functions is equal to the supremum of the biconjugate of the data functions. The study focuses its attention on functions defined in finite-dimensional spaces, in this case the formulae can be simplifi…
In this paper we develop general formulas for the subdifferential of the pointwise supremum of convex functions, which cover and unify both the compact continuous and the non-compact non-continuous settings. From the non-continuous to the continuous setting, we proceed by a compactification-based approach which leads us to problems having compact index sets and upper semi-continuously indexed mappings, giving rise to new characterizations of the subdifferential of the supremum by means of upper semicontinuous regularized functions and an enlarged compact index set. In the opposite sense, we re…
In this paper we develop a geometric approach to convex subdifferential calculus in finite dimensions with employing some ideas of modern variational analysis. This approach allows us to obtain natural and rather easy proofs of basic results of convex subdifferential calculus in full generality and also derive new results of convex analysis concerning marginal/value functions, normal of inverse images of sets under set-valued mappings, calculus rules for coderivatives of single-valued and set-valued mappings, and calculating coderivatives of solution maps to convex generalized equations.
The main contribution of this paper is that every convex function with non-empty relative algebraic interior of its domain is Lipschitz and subdifferentiable in some algebraic sense without any additional topological constraints. The proposed approach uses slightly modified Clarke's subdifferential for functions defined on a convex symmetric set and Lipschitz with respect to a Minkowski functional. Following this, Clarke's subdifferential calculus is generalized to vector spaces and, where continuity properties are claimed, to topological vector spaces.
The paper is devoted to the development of a comprehensive calculus for directional limiting normal cones, subdifferentials and coderivatives in finite dimensions. This calculus encompasses the whole range of the standard generalized differential calculus for (non-directional) limiting notions and relies on very weak (non-restrictive) qualification conditions having also a directional character. The derived rules facilitate the application of tools exploiting the directional limiting notions to difficult problems of variational analysis including, for instance, various stability and sensitivit…
The paper concerns the second-order generalized differentiation theory of variational analysis and new applications of this theory to some problems of constrained optimization in finitedimensional spaces. The main attention is paid to the so-called (full and partial) second-order subdifferentials of extended-real-valued functions, which are dual-type constructions generated by coderivatives of frst-order subdifferential mappings. We develop an extended second-order subdifferential calculus and analyze the basic second-order qualification condition ensuring the fulfillment of the principal seco…
We generalize and improve the original characterization given by Valadier [18, Theorem 1] of the subdifferential of the pointwise supremum of convex functions, involving the subdifferentials of the data functions at nearby points. We remove the continuity assumption made in that work and obtain a general formula for such a subdiferential. In particular, when the supremum is continuous at some point of its domain, but not necessarily at the reference point, we get a simpler version which gives rise to the Valadier formula. Our starting result is the characterization given in [11, Theorem 4], wh…
We generalize and improve the original characterization given by Valadier [20, Theorem 1] of the subdifferential of the pointwise supremum of convex functions, involving the subdifferentials of the data functions at nearby points. We remove the continuity assumption made in that work and obtain a general formula for such a subdifferential. In particular, when the supremum is continuous at some point of its domain, but not necessarily at the reference point, we get a simpler version which gives rise to Valadier formula. Our starting result is the characterization given in [10, Theorem 4], which…
In this paper, in the absence of any constraint qualifications, we develop sequential necessary and sufficient optimality conditions for a constrained multiobjective fractional programming problem characterizing a Henig proper efficient solution in terms of the $ε$-subdifferentials and the subdifferentials of the functions. This is achieved by employing a sequential Henig subdifferential calculus rule of the sums of $m \ (m\geq 2)$ proper convex vector valued mappings with a composition of two convex vector valued mappings. In order to present an example illustrating the main results of this p…
We derive exact calculus rules for the directed subdifferential defined for the class of directed subdifferentiable functions. We also state optimality conditions, a chain rule and a mean-value theorem. Thus we extend the theory of the directed subdifferential from quasidifferentiable to directed subdifferentiable functions.
In this article, we study $gH$-subdifferential calculus of convex interval-valued functions (IVFs) and apply it in a nonconvex composite model of interval optimization problems (IOPs). It is found that the $gH$-directional derivative of maximum of finitely many comparable IVFs is the maximum of their $gH$-directional derivative. Proposed concepts of $gH$-subdifferential are observed to be useful to derive Fritz-John-type and KKT-type efficiency conditions for weak efficient solutions of IOPs. Further, we extract a necessary and sufficient condition to characterize the weak efficient solutions …
Optimization models involving quaternion matrices are widely used in color image process and other engineering areas. These models optimize real functions of quaternion matrix variables. In particular, $\ell_0$-norms and rank functions of quaternion matrices are discrete. Yet calculus with derivatives, subdifferentials and generalized subdifferentials of such real functions is needed to handle such models. In this paper, we introduce first and second order derivatives and establish their calculation rules for such real functions. Our approach is consistent with the subgradient concept for norm…
Construction of an united theory of the subdifferentials of the first and second orders is interesting for many specialists in optimization \cite{morduchrockafel}. In the paper the rules for construction of the subdifferentials of the first and second orders are introduced. The constructions are done with the help of the Steklov integral of a Lipschitz function $f(\cdot)$ over the images of a set-valued mapping $D(\cdot)$. It is proved that the subdifferential of the first order consisting of the average integral limit values of the gradients $\nabla f(r(\cdot))$, calculated along the curves $…
This article provides a definition of a subdifferential for continuous functions based on homological considerations. We show that it satisfies all the requirement for a good notion of subdifferential. Moreover, we prove sublinearity, a Leibniz formula and an approximation result. This work fits in the framework of microlocal analysis of sheaves for C^0 symplectic problems and application to Aubry-Mather theory.
This article introduces the concept of weak sharp minima (WSM) for convex interval-valued functions (IVFs). To identify a set of WSM of a convex IVF, we provide its primal and dual characterizations. The primal characterization is given in terms of $gH$-directional derivatives. On the other hand, to derive dual characterizations, we propose the notions of the support function of a subset of $I(\mathbb{R})^{n}$ and $gH$-subdifferentiability for convex IVFs. Further, we develop the required $gH$-subdifferential calculus for convex IVFs. Thereafter, by using the proposed $gH$-subdifferential calc…
The (delta-) normal cone to an arbitrary intersection of sublevel sets of proper, lower semicontinuous, and convex functions is characterized, using either epsilon-subdifferentials at the nominal point or exact subdifferentials at nearby points. Our tools include (epsilon-) calculus rules for sup/max functions. The framework of this work is that of a locally convex space, however, formulas using exact subdifferentials require some restriction either on the space (e.g. Banach), or on the function (e.g. epi-pointed).
In this paper we study the concept of algebraic core for convex sets in general vector spaces without any topological structure and then present its applications to problems of convex analysis and optimization. Deriving the equivalence between the Hahn-Banach theorem and and a simple version of the separation theorem of convex sets in vector spaces allows us to develop a geometric approach to generalized differential calculus for convex sets, set-valued mappings, and extended-real-valued functions with qualification conditions formulated in terms of algebraic cores for such objects. We also ob…
Using techniques of variational analysis, necessary and sufficient subdifferential conditions for Hölder error bounds are investigated and some new estimates for the corresponding modulus are obtained. As an application, we consider the setting of convex semi-infinite optimization and give a characterization of the Hölder calmness of the argmin mapping in terms of the level set mapping (with respect to the objective function) and a special supremum function. We also estimate the Hölder calmness modulus of the argmin mapping in the framework of linear programming.
We introduce a proximal subdifferential and develop a calculus for nonsmooth functions defined on any Riemannian manifold $M$. We give several applications of this theory, concerning: 1) differentiability and geometrical properties of the distance function to a closed subset $C$ of $M$; 2) solvability and implicit function theorems for nonsmooth functions on $M$; 3) conditions on the existence of a circumcenter for three different points of $M$; and especially 4) fixed point theorems for expansive and nonexpansive mappings and certain perturbations of such mappings defined on $M$.
In this note we provide a full conjugacy and subdifferential calculus for convex convex-composite functions in finite-dimensional space. Our approach, based on infimal convolution and cone-convexity, is straightforward and yields the desired results under a verifiable Slater-type condition, with relaxed monotonicity and without lower semicontinuity assumptions on the functions in play. The versatility of our findings is illustrated by a series of applications in optimization and matrix analysis, including conic programming, matrix-fractional, variational Gram, and spectral functions.
This paper proposes and develops a new Newton-type algorithm to solve subdifferential inclusions defined by subgradients of extended-real-valued prox-regular functions. The proposed algorithm is formulated in terms of the second-order subdifferential of such functions that enjoys extensive calculus rules and can be efficiently computed for broad classes of extended-real-valued functions. Based on this and on metric regularity and subregularity properties of subgradient mappings, we establish verifiable conditions ensuring well-posedness of the proposed algorithm and its local superlinear conve…
The goal of the paper is to study the particular class of regularly ${\mathcal{H}}$-convex functions, when ${\mathcal{H}}$ is the set ${\mathcal{L}\widehat{C}}(X,{\mathbb{R}})$ of real-valued Lipschitz continuous classically concave functions defined on a real normed space $X$. For an extended-real-valued function $f:X \mapsto \overline{\mathbb{R}}$ to be ${\mathcal{L}\widehat{C}}$-convex it is necessary and sufficient that $f$ be lower semicontinuous and bounded from below by a Lipschitz continuous function; moreover, each ${\mathcal{L}\widehat{C}}$-convex function is regularly ${\mathcal{L}\…
Smoothing methods have become part of the standard tool set for the study and solution of nondifferentiable and constrained optimization problems as well as a range of other variational and equilibrium problems. In this note we synthesize and extend recent results due to Beck and Teboulle on infimal convolution smoothing for convex functions with those of X. Chen on gradient consistency for nonconvex functions. We use epi-convergence techniques to define a notion of epi-smoothing that allows us to tap into the rich variational structure of the subdifferential calculus for nonsmooth, nonconvex,…
In the early 1960's, Moreau and Rockafellar introduced a concept of called \emph{subgradient} for convex functions, initiating the developments of theoretical and applied convex analysis. The needs of going beyond convexity motivated the pioneer works by Clarke considering generalized differentiation theory of Lipschitz continuous functions. Although Clarke generalized differentiation theory is applicable for nonconvex functions, convexity still plays a crucial role in Clarke subdifferential calculus. In the mid 1970's, Mordukhovich developed another generalized differentiation theory for nonc…
