Related papers: On the strong concavity of the dual function of an…
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…
The standard assumption for proving linear convergence of first order methods for smooth convex optimization is the strong convexity of the objective function, an assumption which does not hold for many practical applications. In this…
This paper studies hidden convexity properties associated with constrained optimization problems over the set of rotation matrices $\text{SO}(n)$. Such problems are nonconvex due to the constraint $X \in \text{SO}(n)$. Nonetheless, we show…
In this work we present two particular cases of the general duality result for linear optimisation problems over signed measures with infinitely many constraints in the form of integrals of functions with respect to the decision variables…
This work studies the strong duality of non-convex matrix factorization problems: we show that under certain dual conditions, these problems and its dual have the same optimum. This has been well understood for convex optimization, but…
The paper introduces several new concepts for solving nonconvex or nonsmooth optimization problems, including convertible nonconvex function, exact convertible nonconvex function and differentiable convertible nonconvex function. It is…
This paper presents a canonical d.c. (difference of canonical and convex functions) programming problem, which can be used to model general global optimization problems in complex systems. It shows that by using the canonical duality…
Nonconvex optimization is central to modern machine learning, but the general framework of nonconvex optimization yields weak convergence guarantees that are too pessimistic compared to practice. On the other hand, while convexity enables…
With this note we bring again into attention a vector dual problem neglected by the contributions who have recently announced the successful healing of the trouble encountered by the classical duals to the classical linear vector…
We consider scalar equilibrium problems governed by a bifunction in a finite-dimensional framework. By using classical arguments in Convex Analysis, we show that under suitable generalized convexity assumptions imposed on the bifunction,…
This note establishes a limiting formula for the conic Lagrangian dual of a convex infinite optimization problem, correcting the classical version of Karney [Math. Programming 27 (1983) 75-82] for convex semi-infinite programs. A…
This document introduces a strategy to solve linear optimization problems. The strategy is based on the bounding condition each constraint produces on each one of the problem's dimension. The solution of a linear optimization problem is…
We associate with each convex optimization problem, posed on some locally convex space, with infinitely many constraints indexed by the set T, and a given non-empty family H of finite subsets of T, a suitable Lagrangian-Haar dual problem.…
Proximal operators with affine constraints arise in numerous models in nonconvex projection, composite optimization, and structured regularization. However, their efficient computation remains challenging due to the simultaneous presence of…
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…
This paper revisits the well-studied fixed point problem from a unified viewpoint of mathematical modeling and canonical duality theory, i.e. the original problem is first reformulated as a nonconvex optimization problem, its well-posedness…
We consider the problem of analyzing and designing gradient-based discrete-time optimization algorithms for a class of unconstrained optimization problems having strongly convex objective functions with Lipschitz continuous gradient. By…
We study nonconvex quadratic problems (QPs) with quadratic separable constraints, where these constraints can be defined both as inequalities or equalities. We derive sufficient conditions for these types of problems to present the…
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…
Composite functions have been studied for over 40 years and appear in a wide range of optimization problems. Convex analysis of these functions focuses on (i) conditions for convexity of the function based on properties of its components,…