Optimality Conditions for Convex Stochastic Optimization Problems in Banach Spaces with Almost Sure State Constraints

This paper studies constrained optimization problems in Banach spaces without usual differentiability and convexity assumptions on the functionals involved in the data. The aim is to give optimality conditions for the problems from which one can derive the characterization of best approximations. The objective and the inequality constraint functionals are assumed to have one-sided directional derivatives. First-order necessary conditions are given in terms of subdifferentials of the directional derivatives. The notion of “max-pseudoconvexity” weaker than pseudoconvexity is introduced for sufficiency. The optimality conditions are applied to linear and nonlinear Tchebycheff approximation problems to derive the characterization of best approximations.

This study is devoted to constrained optimization problems in Banach spaces. We present different properties of tangent cones associated with an arbitrary subset of a Banach space and establish correlations with some of the existing results. In absence of both differentiability and convexity assumptions on the functions involved in the optimization problem, the consideration of these tangent cones and their polars leads us to introduce new concepts in nondifferentiable programming. Necessary optimality conditions are first developed in a general abstract form; then these conditions are made more precise in the presence of equality constraints by introducing the concept of normal subcone.

We consider pairs of non reflexive Banach spaces $$(E_0, E)$$ such that $$E_0$$ is defined in terms of a little-o condition and E is defined by the corresponding big-O condition. Under suitable assumptions on the pair $$(E_0, E)$$ there exists a reflexive and separable Banach space X (in which E is continuously embedded and dense) naturally associated to E which characterizes quantitatively weak compactness of bounded linear operators $$\begin{aligned} T: E_0 \rightarrow Z \end{aligned}$$ where Z is an arbitrary Banach space. Pairs include (VMO, BMO), where BMO is the space of John-Nirenberg, $$(B_0, B)$$ where B is a recently introduced space by Bourgain-Brezis-Mironescu ([6]) and some Orlicz pairs $$(L^{\psi }_0, L^{\psi })$$ where $$L^{\psi }_0$$ is the closure of $$L^\infty $$ in the Orlicz space $$L^{\psi }$$ , Marcinkiewicz pairs $$(L^{q, \infty }_0, L^{q, \infty })$$ where $$L^{q, \infty }_0$$ is the closure of $$L^\infty $$ in the Marcinkiewicz weak– $$L^q$$ denoted by $$L^{q, \infty }$$ . More generally, Banach function spaces are considered. The main results are duality formulas of the type 1 $$\begin{aligned} E_0^{**}&\simeq E\qquad \text {isometrically}\end{aligned}$$ 2 $$\begin{aligned}E^{*}&\simeq E_0^*\oplus _1 E_0^\perp\end{aligned}$$ and distance formulas.

This paper considers a class of nonlinear differentiable optimization problems depending on a parameter. We show that, if constraint regularity, a second-order sufficient optimality condition, and a stability condition for the Lagrange multipliers hold, then for sufficiently smooth perturbations of the constraints and the objective function the optimal solutions locally obey a type of Lipschitz condition. The results are applied to finite-dimensional problems, equality constrained problems, and optimal control problems.

In this paper we study local behavior of optimal solutions of parametric programs in Banach spaces. A general approach to approximation of the original problem by a simpler one is outlined. Under various regularity conditions, Lipschitz continuity and directional differentiability properties of the optimal solutions are derived.

Function spaces and its various applications are subjects of investigation in many research centers and we are glad that this special issue has received the remarkable attention by researchers in these areas. We have received papers from wide range of topics of function spaces such as interpolation methods for stochastic processes spaces, vector optimization problems in Banach spaces, Marcinkiewicz integral operator in Morrey spaces, Hardy’s operator in the variable exponent Lebesgue spaces, and differential equations of weakly parabolic type in Banach spaces. A number of papers are devoted to investigating different kinds of problems in Hardy spaces or Hardy subspaces, for example, Calderon-Zygumnd singular integral operators in Hardy spaces, weighted Hardy spaces related to the Schrodinger operator, and composition operator in hyperbolic Bloch-type spaces. Geometric problems such as modulus of convexity in some Banach sequence spaces, p-uniform convexity, and quniform smoothness in C spaces are also covered by this issue.

In this paper, the convergence of the fundamental alternating minimization is established for non-smooth non-strongly convex optimization problems in Banach spaces, and novel rates of convergence are provided. As objective function a composition of a smooth, and a block-separable, non-smooth part is considered, covering a large range of applications. For the former, three different relaxations of strong convexity are considered: (i) quasi-strong convexity; (ii) quadratic functional growth; and (iii) plain convexity. With new and improved rates benefiting from both separate steps of the scheme, linear convergence is proved for (i) and (ii), whereas sublinear convergence is showed for (iii).

Let E be a Banach space. The concept of n‐type overE is introduced here, generalizing the concept of type overE introduced by Krivine and Maurey. Let E″ be the second dual of E and fix . The function τ : E × ℝn → ℝ, defined by letting for all x ∈ E and all a1, …, an ∈ ℝ, defines an n‐type over E. Types that can be represented in this way are called double‐dual n‐types; we say that realizes τ. Let E be a (not necessarily separable) Banach space that does not contain ℓ1. We study the set of elements of that realize a given double‐dual n‐type over E. We show that the set of realizations of this n‐type is convex. This generalizes a result of Haydon and Maurey who showed that the set of realizations of a given 1‐type over a separable Banach space E is convex. The proof makes use of Henson′s language for normed space structures and uses ideas from mathematical logic, most notably the Löwenheim‐Skolem theorem.

Weak ∗-Polish Banach spaces

The paper studies weak greedy algorithms for finding sparse solutions of convex optimization problems in Banach spaces. We consider the concept of duality gap, the values of which are implicitly calculated at the step of choosing the direction of the fastest descent at each iteration of the greedy algorithm. We show that these values give upper bounds for the difference between the values of the objective function in the current state and the optimal point. Since the value of the objective function at the optimal point is not known in advance, the current values of the duality gap can be used, for example, in the stopping criteria for the greedy algorithm. In the paper, we find estimates of the duality gap values depending on the number of iterations for the weak greedy algorithms under consideration.

This paper is concerned with generalized differentiation of the efficient point multifunctions of parametric vector optimization problems in Banach spaces. We give formulae for computing and/or estimating the Fréchet coderivative (precoderivative) of this multifunction in a broad class of conventional vector optimization problems with the presence of geometric, operator, (finite and infinite) functional constraints. Examples are given to illustrate the obtained results.

Families of perturbed, (non-convex) optimization problems in Banach spaces have been shown to be densely solvable ; this paper tries to present a unified approach of such density results and highlights their intimate connection with the geometry of Banach spaces.

Every separable Banach space admits an equivalent norm such that the unit ball with respect to this norm has at most countably many strongly extreme points.Every separable nonreflexive Banach space can be renormed so that its unit ball has at most countably many weakly strongly extreme points.

Derivation of necessary conditions for optimality in optimal control theory relies on variational calculus of control systems. We provide here such calculus for the semilinear differential inclusion $$x' \in Ax + F(t,x), x(0) = x_0$$ (1) (1) where A is the infinitesimal generator of a C o-semigroup on a separable Banach space X and F: [0,T]×X∼ X is a set-valued map.

Within the class of reflexive Banach spaces, we prove a metric characterization of the class of asymptotic-c0 spaces in terms of a bi-Lipschitz invariant which involves metrics that generalize the Hamming metric on k-subsets of ℕ. We apply this characterization to show that the class of separable, reflexive, and asymptotic-c0 Banach spaces is non-Borel co-analytic. Finally, we introduce a relaxation of the asymptotic-c0 property, called the asymptotic-subsequential-c0 property, which is a partial obstruction to the equi-coarse embeddability of the sequence of Hamming graphs. We present examples of spaces that are asymptotic-subsequential-c0. In particular, T*(T*) is asymptotic-subsequential-c0 where T* is Tsirelson’s original space.