Subdifferentials Of Convex Functions Research Articles

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.

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

ABSTRACTThe 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.

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.

Weak subdifferential/superdifferential, weak exhausters and optimality conditions

In this work, the notion of weak superdifferential is presented. Some calculation rules are given to evaluate weak subdifferential and weak superdifferential of some classes of functions represented by support functions. Moreover, some methods are obtained to calculate weak subdifferential of convex functions. In addition, the concept of weak lower and weak upper exhausters of positively homogeneous functions are introduced by using weak subdifferential and weak superdifferential, respectively. In terms of weak exhausters, some optimality conditions are given to find local or global minimizers/maximizers of some classes of functions.

From revealed preference to preference revelation

Utility functions are regarded as elements of a linear space that is paired with a dual representation of choices to demonstrate the similarity between preference revelation and the duality of prices and quantities in revealed preference. With respect to preference revelation, quasilinear versus ordinal utility and choices in an abstract set versus choices in a linear space are distinguished and their separate and common features are explored. The central thread uniting the various strands is the subdifferentiability of convex functions.

Subdifferentials of convex functions on time scales

The paper studies the notion of subdifferentials of functions defined on a time scale.The subdifferential of a given function $f$ is defined as the set of certain extended functions.Since the convexity of the given function guarantees its subdifferentiability, properties of convex functions on time scales are presented. We show that the convexity of a function is the necessary and sufficient condition for its subdifferentiability.The relations between the delta, nabla, diamond-$\alpha$ derivatives and subdifferentials ofconvex functions are given.

On a Sufficient Condition for Equality of Two Maximal Monotone Operators

We establish minimal conditions under which two maximal monotone operators coincide. Our first result is inspired by an analogous result for subdifferentials of convex functions. In particular, we prove that two maximal monotone operators T,S which share the same convex-like domain D coincide whenever \(T(x)\cap S(x)\not=\emptyset \) for every x ∈ D. We extend our result to the setting of enlargements of maximal monotone operators. More precisely, we prove that two operators coincide as long as the enlargements have nonempty intersection at each point of their common domain, assumed to be open. We then use this to obtain new facts for convex functions: we show that the difference of two proper lower semicontinuous and convex functions whose subdifferentials have a common open domain is constant if and only if their e-subdifferentials intersect at every point of that domain.

Duality and subdifferential for convex functions on complete [formula omitted] metric spaces

Thanks to the recent concept of quasilinearization of Berg and Nikolaev, we have introduced the notion of duality and subdifferential on complete C A T ( 0 ) (Hadamard) spaces. For a Hadamard space X , its dual is a metric space X ∗ which strictly separates non-empty, disjoint, convex closed subsets of X , provided that one of them is compact. If f : X → ( − ∞ , + ∞ ] is a proper, lower semicontinuous, convex function, then the subdifferential ∂ f : X ⇉ X ∗ is defined as a multivalued monotone operator such that, for any y ∈ X there exists some x ∈ X with x y ⃗ ∈ ∂ f ( x ) . When X is a Hilbert space, it is a classical fact that R ( I + ∂ f ) = X . Using a Fenchel conjugacy-like concept, we show that the approximate subdifferential ∂ ϵ f ( x ) is non-empty, for any ϵ > 0 and any x in efficient domain of f . Our results generalize duality and subdifferential of convex functions in Hilbert spaces.

Nonlinear second order differential inclusions with a series of convex functions

Abstract In this paper, we study a class of second order differential inclusions attached to which the convex potential varies depending on the time, and the boundary conditions are described by the subdifferentials of convex functions. Applying the theory of lower semicontinuous, convex and proper functionals, and fixed points, we give some results about the representation of the subdifferential operators of some convex functionals and the existence of solutions for the boundary value problems.

Restricted Weak Upper Semi-continuity of Subdifferentials of Convex Functions on Banach Spaces

Let X be a Banach space and f a continuous convex function on X. Suppose that for each x ∈ X and each weak neighborhood V of zero in X* there exists δ > 0 such that $$\partial f(y)\subset\partial f(x)+V\;\;{\rm for\;all}\;y\in X\;{\rm with}\;\|y-x\|<\delta. $$ Then every continuous convex function g with \(g \leqslant f\) on X is generically Frechet differentiable. If, in addition, \(\lim\limits_{\|x\|\rightarrow\infty}f(x)=\infty\), then X is an Asplund space.

Natural closures, natural compositions and natural sums of monotone operators

We introduce new methods for defining generalized sums of monotone operators and generalized compositions of monotone operators with linear maps. Under asymptotic conditions we show these operations coincide with the usual ones. When the monotone operators are subdifferentials of convex functions, a similar conclusion holds. We compare these generalized operations with previous constructions by Attouch–Baillon–Théra, Revalski–Théra and Pennanen–Revalski–Théra. The constructions we present are motivated by fuzzy calculus rules in nonsmooth analysis. We also introduce a convergence and a closure operation for operators which may be of independent interest.

Integration of ε-Fenchel Subdifferentials and Maximal Cyclic Monotonicity

This paper concerns the integration of ?-Fenchel subdifferentials of proper lower semicontinuous convex functions defined on arbitrary topological vector spaces. We make use of integration tools to provide a representation formula of the approximate subdifferential of convex functions, and also to identify the class of maximal cyclically monotone families of operators.

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].

Subdifferentials of convex functions and sigma-cyclic monotonicity

The property of σcyclic monotonicity is proposed here to describe subdifferentials of lsc convex functions that are continuous in their domains. It is shown that all monotone operators in R and all densely defined cyclically monotome operators in Rn share this property. Examples of a densely defined maximal cyclically monotone operator in a Hilbert space and of a subdifferential of a convex lsc function in R2 which are not σ-cyclically monotone operators are given.

The maximal monotonicity of the subdifferentials of convex functions: Simons' problem

In the paper we deal with the problem when the graph of the subdifferential operator of a convex lower semicontinuous function has a common point with the product of two convex nonempty weak and weak* compact sets, i.e. when graph ∂ψ ∩ (Q × Q*) ≠ 0. The results obtained partially solve the problem posed by Simons as well as generalize the Rockafellar Maximal Monotonicity Theorem.

On the convergence of subdifferentials of convex functions

On the convergence of subdifferentials of convex functions

Variational formulations of non-linear and non-smooth structural problems

The inverse problem of variational calculus is addressed with reference to structural models governed by non-linear field equations and monotone multi-valued constitutive operators. For such a class of models a non-smooth analysis must be necessarily carried out. The concept of consistency of non-linear strain operators is first recalled in view of a Lagrangian formulation of equilibrium. The structural problem is then recast in terms of a single structural operator which encompasses the field and constitutive equations by means of two sub-operators. The first one, which accounts for equilibrium and compatibility, is proved to be conservative and its potential explicitly derived. The second one is assumed to be conservative since it embodies multi-valued constitutive relations which are expressed as Subdifferentials of convex functionals. The original problem is then amenable to a weak formulation and, recalling recent results on the potential theory of monotone multi-valued operators, a constructive method for the variational formulation of problems expressed in terms of conservative multi-valued operators is presented. The structural operator is accordingly integrated in the product space of all the state variables to get the expression of the associated potential. Further, by enforcing constraint relations and kinematic compatibility, a family of non-smooth func̀tionals is derived and the related stationarity conditions are suitably defined starting from the concept of local subdifferential. Finally, it is shown that the stationarity of each of these functionals yields back an operator form of the structural problem.

$D$-representation of subdifferentials of directionally Lipschitz functions

Subdifferentials of convex functions and some regular functions f are expressed in terms of limiting gradients at points in a given dense subset of dom Vf .

Four inequalities for monotone gradient vector fields

We consider vector fields which have realizations as subdifferentials of convex functionals on a Hilbert space. Four inequalities relating the resolvent of the subdifferential to the semigroup flow are known in the literature. In this paper the constant appearing in one of these inequalities is improved, and the resulting four inequalities are shown to be optimal.

A complement to Trotter's product formula for nonlinear semigroups generated by the subdifferentials of convex functionals

A complement to Trotter's product formula for nonlinear semigroups generated by the subdifferentials of convex functionals