Subdifferential of the supremum function: moving back and forth between continuous and non-continuous settings

In this paper we establish 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 rewrite the subdifferential of these new regularized functions by using the original data, also leading us to new results on the subdifferential of the supremum. We give two applications in the last section, the first one concerning the nonconvex Fenchel duality, and the second one establishing Fritz-John and KKT conditions in convex semi-infinite programming.

     

Integration formulas via the Fenchel subdifferential of nonconvex functions

Starting from explicit expressions for the subdifferential of the conjugate function, we establish in the Banach space setting some integration results for the so-called epi-pointed functions. These results use the ε-subdifferential and the Fenchel subdifferential of an appropriate weak lower semicontinuous (lsc) envelope of the initial function. We apply these integration results to the construction of the lsc convex envelope either in terms of the ε-subdifferential of the nominal function or of the subdifferential of its weak lsc envelope.     

On the Maximization of (not necessarily) Convex Functions on Convex Sets

The global solutions of the problem of maximizing a convex function on a convex set were characterized by several authors using the Fenchel (approximate) subdifferential. When the objective function is quasiconvex it was considered the differentiable case or used the Clarke subdifferential. The aim of the present paper is to give necessary and sufficient optimality conditions using several subdifferentials adequate for quasiconvex functions. In this way we recover almost all the previous results related to such global maximization problems with simple proofs.     

Maximal Monotone Operators, Convex Functions and a Special Family of Enlargements

This work establishes new connections between maximal monotone operators and convex functions. Associated to each maximal monotone operator, there is a family of convex functions, each of which characterizes the operator. The basic tool in our analysis is a family of enlargements, recently introduced by Svaiter. This family of convex functions is in a one-to-one relation with a subfamily of these enlargements. We study the family of convex functions, and determine its extremal elements. An operator closely related to the Legendre–Fenchel conjugacy is introduced and we prove that this family of convex functions is invariant under this operator. The particular case in which the operator is a subdifferential of a convex function is discussed.     

A weaker regularity condition for subdifferential calculus and Fenchel duality in infinite dimensional spaces

In this paper we present a new regularity condition for the subdifferential sum formula of a convex function with the precomposition of another convex function with a continuous linear mapping. This condition is formulated by using the epigraphs of the conjugates of the functions involved and turns out to be weaker than the generalized interior-point regularity conditions given so far in the literature. Moreover, it provides a weak sufficient condition for Fenchel duality regarding convex optimization problems in infinite dimensional spaces. As an application, we discuss the strong conical hull intersection property (CHIP) for a finite family of closed convex sets.     

Calculus rules for global approximate minima and applications to approximate subdifferential calculus

We provide calculus rules for global approximate minima concerning usual operations on functions. The formulas we obtain are then applied to approximate subdifferential calculus. In this way, new results are presented, for example on the approximate subdifferential of a deconvolution, or on the subdifferential of an upper envelope of convex functions.     

Generalized Fenchel’s Conjugation Formulas and Duality for Abstract Convex Functions

In this paper, we present a generalization of Fenchel’s conjugation and derive infimal convolution formulas, duality and subdifferential (and ε-subdifferential) sum formulas for abstract convex functions. The class of abstract convex functions covers very broad classes of nonconvex functions. A nonaffine global support function technique and an extended sum-epiconjugate technique of convex functions play a crucial role in deriving the results for abstract convex functions. An additivity condition involving global support sets serves as a constraint qualification for the duality. Work of Z.Y. Wu was carried out while the author was at the Department of Applied Mathematics, University of New South Wales, Sydney, Australia.     

Duality Principles for Optimization Problems Dealing with the Difference of Vector-Valued Convex Mappings

Using the concept of a subdifferential of a vector-valued convex mapping, we provide duality formulas for the minimization of nonconvex composite functions and related optimization problems such as the minimization of a convex function over a vectorial DC constraint.     

A Duality Theory for Set-Valued Functions I: Fenchel Conjugation Theory

It is proven that a proper closed convex function with values in the power set of a preordered, separated locally convex space is the pointwise supremum of its set-valued affine minorants. A new concept of Legendre–Fenchel conjugates for set-valued functions is introduced and a Moreau–Fenchel theorem is proven. Examples and applications are given, among them a dual representation theorem for set-valued convex risk measures.      

Stable and Total Fenchel Duality for Composed Convex Optimization Problems

In this paper, we consider the composed convex optimization problem which consists in minimizing the sum of a convex function and a convex composite function. By using the properties of the epigraph of the conjugate functions and the subdifferentials of convex functions, we give some new constraint qualifications which completely characterize the strong Fenchel duality and the total Fenchel duality for composed convex optimiztion problem in real locally convex Hausdorff topological vector spaces.     

Symbolic Fenchel Conjugation

Of key importance in convex analysis and optimization is the notion of duality, and in particular that of Fenchel duality. This work explores improvements to existing algorithms for the symbolic calculation of subdifferentials and Fenchel conjugates of convex functions defined on the real line. More importantly, these algorithms are extended to enable the symbolic calculation of Fenchel conjugates on a class of real-valued functions defined on $\mathbb{R}^n$ . These algorithms are realized in the form of the Maple package SCAT.     

Random variables,monotone relations,and convex analysis

Random variables can be described by their cumulative distribution functions, a class of nondecreasing functions on the real line. Those functions can in turn be identified, after the possible vertical gaps in their graphs are filled in, with maximal monotone relations. Such relations are known to be the subdifferentials of convex functions. Analysis of these connections yields new insights. The generalized inversion operation between distribution functions and quantile functions corresponds to graphical inversion of monotone relations. In subdifferential terms, it corresponds to passing to conjugate convex functions under the Legendre–Fenchel transform. Among other things, this shows that convergence in distribution for sequences of random variables is equivalent to graphical convergence of the monotone relations and epigraphical convergence of the associated convex functions. Measures of risk that employ quantiles (VaR) and superquantiles (CVaR), either individually or in mixtures, are illuminated in this way. Formulas for their calculation are seen from a perspective that reveals how they were discovered. The approach leads further to developments in which the superquantiles for a given distribution are interpreted as the quantiles for an overlying “superdistribution.” In this way a generalization of Koenker–Basset error is derived which lays a foundation for superquantile regression as a higher-order extension of quantile regression.     

Some conjugation formulas and subdifferential formulas of convex analysis revisited

We give simpler proofs of some known conjugation formulas and subdifferential formulas of convex analysis and we give some new interconnections between them, showing how each of them follows from the others.     

Differential analysis of polarity: Polar Hamilton-Jacobi,conservation laws,and Monge Ampère equations

We develop a differential theory for the polarity transform parallel to that of the Legendre transform, which is applicable when the functions studied are “geometric convex”, namely, convex, non-negative, and vanish at the origin. This analysis establishes basic tools for dealing with this duality transform, such as the polar subdifferential map, and variational formulas. Another crucial step is identifying a new, non-trivial, sub-class of C ² functions preserved under this transform. This analysis leads to a new method for solving many new first order equations reminiscent of Hamilton–Jacobi and conservation law equations, as well as some second order equations of Monge–Ampère type. This article develops the theory of strong solutions for these equations which, due to the nonlinear nature of the polarity transform, is considerably more delicate than its counterparts involving the Legendre transform. As one application, we introduce a polar form of the homogeneous Monge–Ampère equation that gives a dynamical meaning to a new method of interpolating between convex functions and bodies. A number of other applications, e.g., to optimal transport and affine differential geometry are considered in sequels.     

Minimal convex majorants of functions and Demyanov–Rubinov exhaustive super(sub)differentials

ABSTRACT

The primary goal of the paper is to establish characteristic properties of (extended) real-valued functions defined on normed vector spaces that admit the representation as the lower envelope (the pointwise infimum) of their minimal (with respect of the pointwise ordering) convex majorants. The results presented in the paper generalize and extend the well-known Demyanov-Rubinov characterization of upper semicontinuous positively homogeneous functions as the lower envelope of exhaustive families of continuous sublinear functions to larger classes of (not necessarily positively homogeneous) functions defined on arbitrary normed spaces. As applications of the above results, we introduce, for nonsmooth functions, a new notion of the Demyanov-Rubinov exhaustive subdifferential at a given point, and show that it generalizes a number of known notions of subdifferentiability, in particular, the Fenchel-Moreau subdifferential of convex functions, the Dini-Hadamard (directional) subdifferential of directionally differentiable functions, and the Φ-subdifferential in the sense of the abstract convexity theory. Some applications of Demyanov-Rubinov exhaustive subdifferentials to extremal problems are considered.     

Subdifferential estimate of the directional derivative,optimality criterion and separation principles

We provide an inequality relating the radial directional derivative and the subdifferential of proper lower semicontinuous functions, which extends the known formula for convex functions. We show that this property is equivalent to other subdifferential properties of Banach spaces, such as controlled dense subdifferentiability, optimality criterion, mean value inequality and separation principles. As an application, we obtain a first-order sufficient condition for optimality, which extends the known condition for differentiable functions in finite-dimensional spaces and which amounts to the maximal monotonicity of the subdifferential for convex lower semicontinuous functions. Finally, we establish a formula describing the subdifferential of the sum of a convex lower semicontinuous function with a convex inf-compact function in terms of the sum of their approximate ?-subdifferentials. Such a formula directly leads to the known formula relating the directional derivative of a convex lower semicontinuous function to its approximate ?-subdifferential.     

Maximal Abstract Monotonicity and Generalized Fenchel’s Conjugation Formulas

In this paper, we present a generalization of a theory of monotone operators in the framework of abstract convexity. We show that how generalized Fenchel’s conjugation formulas can be used to obtain some results on maximal abstract monotonicity. We give a necessary and sufficient condition for maximality of abstract monotone operators representable by abstract convex functions by using an additivity constraint qualification.     

A subdifferential characterization of Motzkin decomposable functions

Abstract

The paper provides a new subdifferential characterization for Motzkin decomposable (convex) functions. This characterization leads to diverse stability properties for such a decomposability for operations like addition and composition.     

Infimal convolution, c-subdifferentiability, and Fenchel duality in evenly convex optimization

In this paper we deal with strong Fenchel duality for infinite-dimensional optimization problems where both feasible set and objective function are evenly convex. To this aim, via perturbation approach, a conjugation scheme for evenly convex functions, based on generalized convex conjugation, is used. The key is to extend some well-known results from convex analysis, involving the sum of the epigraphs of two conjugate functions, the infimal convolution and the sum formula of ??-subdifferentials for lower semicontinuous convex functions, to this more general framework.     

Subgradients of value functions in parametric dynamic programming

We study in this paper the first-order behavior of value functions in parametric dynamic programming with linear constraints and nonconvex cost functions. By establishing an abstract result on the Fréchet subdifferential of value functions of parametric mathematical programming problems, some new formulas on the Fréchet subdifferential of value functions in parametric dynamic programming are obtained.