Related papers: Convex analysis for composite functions without K-…
We review several (and provide new) results on the theory of moments, sums of squares and basic semi-algebraic sets when convexity is present. In particular, we show that under convexity, the hierarchy of semidefinite relaxations for…
This article discusses nonconforming finite element methods for convex minimization problems and systematically derives dual mixed formulations. Duality relations lead to simple error estimates that avoid an explicit treatment of…
Discrete convex functions are used in many areas, including operations research, discrete-event systems, game theory, and economics. The objective of this paper is to offer a survey on fundamental operations for various kinds of discrete…
We extend the standard notion of self-concordance to non-convex optimization and develop a family of second-order algorithms with global convergence guarantees. In particular, two function classes -- \textit{weakly self-concordant}…
In this paper, we propose a new Fully Composite Formulation of convex optimization problems. It includes, as a particular case, the problems with functional constraints, max-type minimization problems, and problems of Composite…
The theory of abstract convexity, also known as convexity without linearity, is an extension of the classical convex analysis. There are a number of remarkable results, mostly concerning duality, and some numerical methods, however, this…
This paper explores optimality conditions in optimization problems involving generalized invex fuzzy functions. We extend the classical KKT framework to settings in which the objective and constraint functions are nonsmooth, vector-valued,…
Convexity is an important notion in non linear optimization theory as well as in infinite dimensional functional analysis. As will be seen below, very simple and powerful tools will be derived from elementary duality arguments (which are…
Motivated by modern regression applications, in this paper, we study the convexification of a class of convex optimization problems with indicator variables and combinatorial constraints on the indicators. Unlike most of the previous work…
In discrete convex analysis, various convexity concepts are considered for discrete functions such as separable convexity, L-convexity, M-convexity, integral convexity, and multimodularity. These concepts of discrete convex functions are…
In this article we utilise abstract convexity theory in order to unify and generalize many different concepts from nonsmooth analysis. We introduce the concepts of abstract codifferentiability, abstract quasidifferentiability and abstract…
We revisit the classical dual ascent algorithm for minimization of convex functionals in the presence of linear constraints, and give convergence results which apply even for non-convex functionals. We describe limit points in terms of the…
In this article we develop a duality principle suitable for a large class of problems in optimization. The main result is obtained through basic tools of convex analysis and duality theory. We establish a correct relation between the…
This paper proposes a general duality framework for the problem of minimizing a convex integral functional over a space of stochastic processes adapted to a given filtration. The framework unifies many well-known duality frameworks from…
This study focuses on convex functions and their generalized. Thus, we start this study by giving the definition of convex functions and some of their properties and discussing a simple geometric property. Then we generalize E-convex…
We study conjugate and Lagrange dualities for composite optimization problems within the framework of abstract convexity. We provide conditions for zero duality gap in conjugate duality. For Lagrange duality, intersection property is…
This paper studies duality and optimality conditions for general convex stochastic optimization problems. The main result gives sufficient conditions for the absence of a duality gap and the existence of dual solutions in a locally convex…
Functions that are piecewise defined are a common sight in mathematics while convexity is a property especially desired in optimization. Suppose now a piecewise-defined function is convex on each of its defining components - when can we…
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…
In this paper we consider convex optimization problems with stochastic composite objective function subject to (possibly) infinite intersection of constraints. The objective function is expressed in terms of expectation operator over a sum…