Related papers: Classical linear vector optimization duality revis…
We show that a wide class of risk-constrained nonconvex functional optimization problems exhibit strong duality, regardless of nonconvexity. We develop two novel results under distinct sets of assumptions, establishing strong duality over…
We develop a methodology for closing duality gap and guaranteeing strong duality in infinite convex optimization. Specifically, we examine two new Lagrangian-type dual formulations involving infinitely many dual variables and infinite sums…
We distinguish two kinds of piecewise linear functions and provide an interesting representation for a piecewise linear function between two normed spaces. Based on such a representation, we study a fully piecewise linear vector…
This paper discusses a special kind of convex constrained optimization problem, whose constraints consist of box inequalities and linear equalities. For this problem, in addition to general optimization algorithms such as exact penalty…
Duality theorems play a fundamental role in convex optimization. Recently, it was shown how duality theorems for countable probability distributions and finite-dimensional quantum states can be leveraged for building relatively complete…
Classification is a common AI problem, and vector search is a typical solution. This transforms a given body of text into a numerical representation, known as an embedding, and modern improvements to vector search focus on optimising speed…
The problem of finding the sparsest vector (direction) in a low dimensional subspace can be considered as a homogeneous variant of the sparse recovery problem, which finds applications in robust subspace recovery, dictionary learning,…
Weak optimal transport generalizes the classical theory of optimal transportation to nonlinear cost functions and covers a range of problems that lie beyond the traditional theory - including entropic transport, martingale transport, and…
In this paper, vector optimization is considered in the framework of decision making and optimization in general spaces. Interdependencies between domination structures in decision making and domination sets in vector optimization are…
The integration of optimization problems within neural network architectures represents a fundamental shift from traditional approaches to handling constraints in deep learning. While it is long known that neural networks can incorporate…
Autonomous agents must often deal with conflicting requirements, such as completing tasks using the least amount of time/energy, learning multiple tasks, or dealing with multiple opponents. In the context of reinforcement learning~(RL),…
This paper studies the distributed optimization problem with possibly nonidentical local constraints, where its global objective function is composed of $N$ convex functions. The aim is to solve the considered optimization problem in a…
This paper addresses the study of algebraic versions of Farkas lemma and strong duality results in the very broad setting of infinite-dimensional conic linear programming in dual pairs of vector spaces. To this end, purely algebraic…
We study the conditions under which the convex relaxation of a mixed-integer linear programming formulation for ordered optimization problems, where sorting is part of the decision process, yields integral optimal solutions. Thereby solving…
Originally designed for applications in computer graphics, visual computing (VC) methods synthesize information about physical and virtual worlds, using prescribed algorithms optimized for spatial computing. VC is used to analyze geometry,…
In this paper we consider distributed optimization problems in which the cost function is separable, i.e., a sum of possibly non-smooth functions all sharing a common variable, and can be split into a strongly convex term and a convex one.…
We present a new kind of Lagrangian duality theory for set-valued convex optimization problems whose objective and constraint maps are defined between preordered normed spaces. The theory is accomplished by introducing a new set-valued…
Non-convex optimization plays a key role in a growing number of machine learning applications. This motivates the identification of specialized structure that enables sharper theoretical analysis. One such identified structure is…
We study dual-based algorithms for distributed convex optimization problems over networks, where the objective is to minimize a sum $\sum_{i=1}^{m}f_i(z)$ of functions over in a network. We provide complexity bounds for four different…
In this thesis, we present results related to complementarity problems. We study the linear complementarity problems on extended second order cones. We convert a linear complementarity problem on an extended second order cone into a mixed…