Subdifferential of the closed convex hull of a function and integration with nonconvex data in general normed spaces

In this paper we approach the study of the subdifferential of the closed convex hull of a function and the related integration problem without the usual assumption of epi-pointedness. The key tool is, as in Hiriart-Urruty et al. (2011) [7], the concept of ε-subdifferential. Some other assumptions which are standard in the literature are also removed.     

<Emphasis Type="Italic">ε</Emphasis>-subgradient algorithms for locally lipschitz functions on Riemannian manifolds

This paper presents a descent direction method for finding extrema of locally Lipschitz functions defined on Riemannian manifolds. To this end we define a set-valued mapping \(x\rightarrow \partial _{\varepsilon } f(x)\) named ε-subdifferential which is an approximation for the Clarke subdifferential and which generalizes the Goldstein- ε-subdifferential to the Riemannian setting. Using this notion we construct a steepest descent method where the descent directions are computed by a computable approximation of the ε-subdifferential. We establish the global convergence of our algorithm to a stationary point. Numerical experiments illustrate our results.     

On the Dini-Hadamard subdifferential of the difference of two functions

In this paper we first provide a general formula of inclusion for the Dini-Hadamard ε-subdifferential of the difference of two functions and show that it becomes equality in case the functions are directionally approximately starshaped at a given point and a weak topological assumption is fulfilled. To this end we give a useful characterization of the Dini-Hadamard ε-subdifferential by means of sponges. The achieved results are employed in the formulation of optimality conditions via the Dini-Hadamard subdifferential for cone-constrained optimization problems having the difference of two functions as objective.     

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.     

Q-subdifferential and Q-conjugate for global optimality

Normal cone and subdifferential have been generalized through various continuous functions; in this article, we focus on a non separable Q-subdifferential version. Necessary and sufficient optimality conditions for unconstrained nonconvex problems are revisited accordingly. For inequality constrained problems, Q-subdifferential and the lagrangian multipliers, enhanced as continuous functions instead of scalars, allow us to derive new necessary and sufficient optimality conditions. In the same way, the Legendre-Fenchel conjugate is generalized into Q-conjugate and global optimality conditions are derived by Q-conjugate as well, leading to a tighter inequality.     

Visualization of the $$\varepsilon $$-subdifferential of piecewise linear–quadratic functions

Computing explicitly the \(\varepsilon \)-subdifferential of a proper function amounts to computing the level set of a convex function namely the conjugate minus a linear function. The resulting theoretical algorithm is applied to the the class of (convex univariate) piecewise linear–quadratic functions for which existing numerical libraries allow practical computations. We visualize the results in a primal, dual, and subdifferential views through several numerical examples. We also provide a visualization of the Brøndsted–Rockafellar theorem.     

Subdifferential properties for a class of minimal time functions with moving target sets in normed spaces

In a normed vector space, we study the minimal time function determined by a moving target set and a differential inclusion, where the set-valued mapping involved has constant values of a bounded closed convex set U. After establishing a characterization of ?-subdifferential of the minimal time function, we obtain that the limiting subdifferential of the minimal time function is representable by virtue of the corresponding normal cones of sublevel sets of the function and level or sublevel sets of the support function of U. The known results require the set U to have the origin as an interior point and the target set is a fixed set.     

An Application of the Bivariate Inf-Convolution Formula to Enlargements of Monotone Operators

Motivated by a classical result concerning the ε-subdifferential of the sum of two proper, convex and lower semicontinuous functions, we give in this paper a similar result for the enlargement of the sum of two maximal monotone operators defined on a Banach space. This is done by establishing a necessary and sufficient condition for a bivariate inf-convolution formula.     

Characterizations of the solution set for non-essentially quasiconvex programming

Characterizations of the solution set in terms of subdifferentials play an important role in research of mathematical programming. Previous characterizations are based on necessary and sufficient optimality conditions and invariance properties of subdifferentials. Recently, characterizations of the solution set for essentially quasiconvex programming in terms of Greenberg–Pierskalla subdifferential are studied by the authors. Unfortunately, there are some examples such that these characterizations do not hold for non-essentially quasiconvex programming. As far as we know, characterizations of the solution set for non-essentially quasiconvex programming have not been studied yet. In this paper, we study characterizations of the solution set in terms of subdifferentials for non-essentially quasiconvex programming. For this purpose, we use Martínez–Legaz subdifferential which is introduced by Martínez–Legaz as a special case of c-subdifferential by Moreau. We derive necessary and sufficient optimality conditions for quasiconvex programming by means of Martínez–Legaz subdifferential, and, as a consequence, investigate characterizations of the solution set in terms of Martínez–Legaz subdifferential. In addition, we compare our results with previous ones. We show an invariance property of Greenberg–Pierskalla subdifferential as a consequence of an invariance property of Martínez–Legaz subdifferential. We give characterizations of the solution set for essentially quasiconvex programming in terms of Martínez–Legaz subdifferential.     

Subdifferentials of multifunctions and Lagrange multipliers for multiobjective optimization

This paper investigates vector optimization problems with objective and the constraints are multifunctions. By using a special scalarization function introduced in optimization by Hiriart-Urruty, we establish optimality conditions in terms of Lagrange-Fritz-John and Lagrange-Kuhn-Tucker multipliers. When all the data of the problem are subconvexlike we derive the results by Li, and hence those of Lin and Corley. We also show how the generalized Moreau-Rockafellar type theorem to multifunctions obtained recently by Lin can be derived from the well-known results in scalar optimization. In the last, vector optimization problem in which objective and the constraints are defined by multifunctions and depends on a parameter u, and the resulting value multifunction M(u) are considered. With the help of the generalized Moreau-Rockafellar type theorem we establish the weak subdifferential of M in terms of the weak subdifferential of objective and constraint multifunctions.     

ε-Subdifferentials in terms of subdifferentials

In this note, we give a formula which expresses the ε-subdifferential operator of a lower semicontinuous convex proper function on a given Banach space in terms of its subdifferential.     

On partial regularity for weak solutions to the Navier-Stokes equations

In this paper, we study the partial regularity of the general weak solution u∈L∞(0,T;L2(Ω))∩L2(0,T;H1(Ω)) to the Navier-Stokes equations, which include the well-known Leray-Hopf weak solutions. It is shown that there is a absolute constant ε such that for the weak solution u, if either the scaled local L^q(1?q?2) norm of the gradient of the solution, or the scaled local ) norm of u is less than ε, then u is locally bounded. This implies that the one-dimensional Hausdorff measure is zero for the possible singular point set, which extends the corresponding result due to Caffarelli et al. (Comm. Pure Appl. Math. 35 (1982) 717) to more general weak solution.     

Sensitivity analysis of nondifferentiable sums of singular values of rectangular matrices

Let K(x) be an s-by-r rectangular matrix depending on a parameter x ε E and denote by g(x) the sum of its m largest singular values (1 ≤ m ≤ Min{s,r}). If K(x) depends affinely on x, then g is a nondifferentiable convex function. In this paper we consider first the affine case and give some formulas for the conjugate, subdifferential, and ε-subdifferential of g. These formulas are then used to obtain perturbation bounds for g(x). We study next the nonaffine case and discuss some questions related with the regularity, generalized subdifferentiability, and directional differentiability of g.     

Remarks on subgradients and ?-subgradients

We show that Rockafellar's maximal monotonicity and maximal cyclical monotonicity theorems for subdifferentials can be reformulated and proved for the family of -subdifferentials of a proper, lower semicontinuous, convex function defined on a normed space. We also show that the subdifferential map of a lower semicontinuous convex function defined on a Banach space is bothX andX* locally maximal monotone.     

Interface boundary value problem for the Navier-Stokes equations in thin two-layer domains

We study a system of 3D Navier-Stokes equations in a two-layer parallelepiped-like domain with an interface coupling of the velocities and mixed (free/periodic) boundary condition on the external boundary. The system under consideration can be viewed as a simplified model describing some features of the mesoscale interaction of the ocean and atmosphere. In case when our domain is thin (of order ε), we prove the global existence of the strong solutions corresponding to a large set of initial data and forcing terms (roughly, of order ε^−2/3). We also give some results concerning the large time dynamics of the solutions. In particular, we prove a spatial regularity of the global weak attractor.     

Unavoidable subgraphs of colored graphs

A natural generalization of graph Ramsey theory is the study of unavoidable sub-graphs in large colored graphs. In this paper, we find a minimal family of unavoidable graphs in two-edge-colored graphs. Namely, for a positive even integer k, let S_k be the family of two-edge-colored graphs on k vertices such that one of the colors forms either two disjoint K_k/2's or simply one K_k/2. Bollobás conjectured that for all k and ε>0, there exists an n(k,ε) such that if n?n(k,ε) then every two-edge-coloring of K_n, in which the density of each color is at least ε, contains a member of this family. We solve this conjecture and present a series of results bounding n(k,ε) for different ranges of ε. In particular, if ε is sufficiently close to , the gap between our upper and lower bounds for n(k,ε) is smaller than those for the classical Ramsey number R(k,k).     

A characterisation of minimal subdifferential mappings of locally Lipschitz functions

In this paper we characterise, in terms of the upper Dini derivative, the Clarke subdifferential mapping being a minimal weak* cusco, and we show that on any Banach space the Clarke subdifferential mapping of a pseudo-regular or semi-smooth locally Lipschitz function is always a minimal weak* cusco.     

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 (Q × Q ^*) 0. The results obtained partially solve the problem posed by Simons as well as generalize the Rockafellar Maximal Monotonicity Theorem.     

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.     

Analytically heavy spaces: Analytic Cantor and Analytic Baire Theorems

Motivated by recent work, we establish the Baire Theorem in the broad context afforded by weak forms of completeness implied by analyticity and K-analyticity, thereby adding to the ‘Baire space recognition literature’ (cf. Aarts and Lutzer (1974) [1], Haworth and McCoy (1977) [43]). We extend a metric result of van Mill, obtaining a generalization of Oxtoby's weak α-favourability conditions (and therefrom variants of the Baire Theorem), in a form in which the principal role is played by K-analytic (in particular analytic) sets that are ‘heavy’ (everywhere large in the sense of some σ-ideal). From this perspective fine-topology versions are derived, allowing a unified view of the Baire Theorem which embraces classical as well as generalized Gandy-Harrington topologies (including the Ellentuck topology), and also various separation theorems. A multiple-target form of the Choquet Banach-Mazur game is a primary tool, the key to which is a restatement of the Cantor Theorem, again in K-analytic form.