Related papers: On some global implicit function theorems for set-…

Set- and vector-valued optimization problems can be re-formulated as complete lattice-valued problems. This has several advantages, one of which is the existence of a clear-cut solution concept which includes the attainment as the infimum…

Local solutions for variational and quasi-variational inequalities are usually the best type of solutions that could practically be obtained when in case of lack of convexity or else when available numerical techniques are too limited for…

For multi-valued functions---such as when the conditional distribution on targets given the inputs is multi-modal---standard regression approaches are not always desirable because they provide the conditional mean. Modal regression…

In this paper, we provide a mathematical framework for improving generalization in a class of learning problems which is related to point estimations for modeling of high-dimensional nonlinear functions. In particular, we consider a…

Variational analysis provides the theoretical foundations and practical tools for constructing optimization algorithms without being restricted to smooth or convex problems. We survey the central concepts in the context of a concrete but…

Multi-class classification problem is among the most popular and well-studied statistical frameworks. Modern multi-class datasets can be extremely ambiguous and single-output predictions fail to deliver satisfactory performance. By allowing…

This paper explores some sufficient conditions for the enhanced solvability of strong vector equilibrium problems, which can be established via a variational approach. Enhanced solvability here means existence of solutions, which are strong…

Extended real-valued functions are often used in optimization theory, but in different ways for infimum problems and for supremum problems. We present an approach to extended real-valued functions that works for all types of problems and…

The paper is devoted to the existence of global optimal solutions for a general class of nonsmooth problems of constrained vector optimization without boundedness assumptions on constraint sets. The main attention is paid to the two major…

In this article, we work with set-valued optimization problems in locally convex topological vector spaces. We prove the equivalencies of some definitions of generalized convex maps introduced by Jeyakumar, Yang, Yang & Yang & Chen, as well…

In many naturally occurring optimization problems one needs to ensure that the definition of the optimization problem lends itself to solutions that are tractable to compute. In cases where exact solutions cannot be computed tractably, it…

We consider the problem of maximizing an unknown function over a compact and convex set using as few observations as possible. We observe that the optimization of the function essentially relies on learning the induced bipartite ranking…

Set-valued prediction is a well-known concept in multi-class classification. When a classifier is uncertain about the class label for a test instance, it can predict a set of classes instead of a single class. In this paper, we focus on…

Regularization plays a key role in a variety of optimization formulations of inverse problems. A recurring theme in regularization approaches is the selection of regularization parameters, and their effect on the solution and on the optimal…

We consider a class of infinite-dimensional optimization problems in which a distributed vector-valued variable should pointwise almost everywhere take values from a given finite set $\mathcal{M}\subset\mathbb{R}^m$. Such hybrid…

Consider a convex function that is invariant under an group of transformations. If it has a minimizer, does it also have an invariant minimizer? Variants of this problem appear in nonparametric statistics and in a number of adjacent fields.…

This works aims at understanding further convergence properties of first order local search methods with complex geometries. We focus on the composite optimization model which unifies within a simple formalism many problems of this type. We…

In this paper we consider the problem of optimization of approximate integration of set-valued functions from the class defined by given majorant of their moduli of continuity, using values of the functions at $n$ fixed or free points of…

The development and identification of effective optimization algorithms for non-convex real-world problems is a challenge in global optimization. Because theoretical performance analysis is difficult, and problems based on models of…

Implicit variables of a mathematical program are variables which do not need to be optimized but are used to model feasibility conditions. They frequently appear in several different problem classes of optimization theory comprising bilevel…