Related papers: Duality of nonconvex optimization with positively …
We prove sufficient and necessary conditions ensuring zero duality gap for Lagrangian duality in some classes of nonconvex optimization problems. To this aim, we use the $\Phi$-convexity theory and minimax theorems for $\Phi$-convex…
Multiobjective integer programs (MOIPs) simultaneously optimize multiple objective functions over a set of linear constraints and integer variables. In this paper, we present continuous, convex hull and Lagrangian relaxations for MOIPs and…
This article explores distributed convex optimization with globally-coupled constraints, where the objective function is a general nonsmooth convex function, the constraints include nonlinear inequalities and affine equalities, and the…
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…
We consider the problem of globally minimizing the sum of many rational functions over a given compact semialgebraic set. The number of terms can be large (10 to 100), the degree of each term should be small (up to 10), and the number of…
Signal processing is rich in inherently continuous and often nonlinear applications, such as spectral estimation, optical imaging, and super-resolution microscopy, in which sparsity plays a key role in obtaining state-of-the-art results.…
In this note we explore duality in reverse convex optimization with reverse convex inequality constraints. While we are examining the special case of a finite index set of the inequality constraints, we are primarily interested in the…
We propose an inexact proximal augmented Lagrangian framework with explicit inner problem termination rule for composite convex optimization problems. We consider arbitrary linearly convergent inner solver including in particular stochastic…
We propose an adaptive accelerated smoothing technique for a nonsmooth convex optimization problem where the smoothing update rule is coupled with the momentum parameter. We also extend the setting to the case where the objective function…
We prove a strengthened form of convexity for operator monotone decreasing positive functions defined on the positive real numbers. This extends Ando and Hiai's work to allow arbitrary positive maps instead of states (or the identity map),…
The aim of this manuscript is to approach by means of first order differential equations/inclusions convex programming problems with two-block separable linear constraints and objectives, whereby (at least) one of the components of the…
Single-level reformulations of (non-convex) distributionally robust optimization (DRO) problems are often intractable, as they contain semiinfinite dual constraints. Based on such a semiinfinite reformulation, we present a safe…
This work presents a unified framework that combines global approximations with locally built models to handle challenging nonconvex and nonsmooth composite optimization problems, including cases involving extended real-valued functions. We…
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…
Generalized polyhedral convex optimization problems in locally convex Hausdorff topological vector spaces are studied systematically in this paper. We establish solution existence theorems, necessary and sufficient optimality conditions,…
Strong (Lagrangian) duality of general conic optimization problems (COPs) has long been studied and its profound and complicated results appear in different forms in a wide range of literatures. As a result, characterizing the known and…
Many problems in nonlinear analysis and optimization, among them variational inequalities and minimization of convex functions, can be reduced to finding zeros (namely, roots) of set-valued operators. Hence numerous algorithms have been…
Many optimization problems arising in high-dimensional statistics decompose naturally into a sum of several terms, where the individual terms are relatively simple but the composite objective function can only be optimized with iterative…
In this work, we revisit a classical incremental implementation of the primal-descent dual-ascent gradient method used for the solution of equality constrained optimization problems. We provide a short proof that establishes the linear…
Optimization methods are at the core of many problems in signal/image processing, computer vision, and machine learning. For a long time, it has been recognized that looking at the dual of an optimization problem may drastically simplify…