CODERIVATIVES OF EFFICIENT POINT MULTIFUNCTIONS IN PARAMETRIC VECTOR OPTIMIZATION

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 present paper, an analysis of the stability behaviour of ideal efficient solutions to parametric vector optimization problems is conducted. A sufficient condition for the existence of ideal efficient solutions to locally perturbed problems and their nearness to a given reference value is provided by refining recent results on the stability theory of parameterized set-valued inclusions. More precisely, the Lipschitz lower semicontinuity property of the solution mapping is established, with an estimate of the related modulus. A notable consequence of this fact is the calmness behaviour of the ideal value mapping associated to the parametric class of vector optimization problems. Within such an analysis, a refinement of a recent existence result, specific for ideal efficient solutions to unperturbed problem and enhanced by related error bounds, is discussed. Some connections with the concept of robustness in multi-objective optimization are also sketched.

In this paper, we deal with the sensitivity analysis in vector optimization. More specifically, formulae for inner and outer evaluating the S-derivative of the efficient point multifunction in parametric vector optimization problems are established. These estimating formulae are presented via the set of efficient/weakly efficient points of the S-derivative of the original multifunction, a composite multifunction of the objective function and the constraint mapping. The elaboration of the formulae in vector optimization problems, having multifunction constraints and semiinfinite constraints, is also undertaken. Furthermore, examples are provided for analyzing and illustrating the obtained results.

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.

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).

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 paper is concerned with higher-order sensitivity analysis in parametric vector optimization problems. Firstly, higher-order proto-differentiability of a set-valued mapping from one Euclidean space to another is defined. Then, we prove that the perturbation map/the proper perturbation map/the weak perturbation map of a parameterized vector optimization problem are higher-order proto-differentiable under some suitable qualification conditions.

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 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 analyze a convex stochastic optimization problem where the state is\nassumed to belong to the Bochner space of essentially bounded random variables\nwith images in a reflexive and separable Banach space. For this problem, we\nobtain optimality conditions that are, with an appropriate model, necessary and\nsufficient. Additionally, the Lagrange multipliers associated with optimality\nconditions are integrable vector-valued functions and not only measures. A\nmodel problem is given demonstrating the application to PDE-constrained\noptimization under uncertainty with an outlook for further applications.\n

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.

Characterization of vector strict global minimizers of order 2 in differentiable vector optimization problems under a new approximation method

Recently some specific classes of non-smooth and non-Lipsch-itz convex optimization problems were considered by Yu. Nesterov and H. Lu. We consider convex programming problems with similar smoothness conditions for the objective function and functional constraints. We introduce a new concept of an inexact model and propose some analogues of switching subgradient schemes for convex programming problems for the relatively Lipschitz-continuous objective function and functional constraints. Some class of online convex optimization problems is considered. The proposed methods are optimal in the class of optimization problems with relatively Lipschitz-continuous objective and functional constraints.

This paper aims at investigating the stability of minimal solutions to set optimization problems on Banach lattices. We introduce the concepts of strictly quasi cone-convexlikeness and supremum mapping of the set-valued mappings and obtain an existence theorem of minimal solutions to set optimization problems by using Zorn's lemma. We first derive the continuity of supremum mapping of the set-valued mapping. Then, we establish the upper semicontinuity and lower semicontinuity of the minimal solution mapping and weak minimal solution mapping to parametric set optimization problems by utilizing strictly quasi cone-convexlikeness and the continuity of supremum mapping. Finally, our main results are applied to semicontinuity of the solution mappings to parametric vector optimization problems.