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We consider the problem of approximating the solution of variational problems subject to the constraint that the admissible functions must be convex. This problem is at the interface between convex analysis, convex optimization, variational…
Given the superposition of a low-rank matrix plus the product of a known fat compression matrix times a sparse matrix, the goal of this paper is to establish deterministic conditions under which exact recovery of the low-rank and sparse…
This paper considers stochastic optimization problems with weakly convex objective and constraint functions. We propose Prox-PEP, a proximal method equipped with quadratic subproblems. To handle nonlinear equality constraints, we employ an…
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…
In multi-objective optimization, computing the entire non-dominated set (also known as the Pareto front or the Pareto frontier) is often intractable. However, for any multiplicative factor greater than one, an approximation set can be…
The popular Lasso approach for sparse estimation can be derived via marginalization of a joint density associated with a particular stochastic model. A different marginalization of the same probabilistic model leads to a different…
We consider several classes of highly important semidefinite optimization problems that involve both a convex objective function (smooth or nonsmooth) and additional linear or nonlinear smooth and convex constraints, which are ubiquitous in…
Graphs are naturally sparse objects that are used to study many problems involving networks, for example, distributed learning and graph signal processing. In some cases, the graph is not given, but must be learned from the problem and…
In this paper, we investigate optimization problems with nonnegative and orthogonal constraints, where any feasible matrix of size $n \times p$ exhibits a sparsity pattern such that each row accommodates at most one nonzero entry. Our…
Decomposition techniques for linear programming are difficult to extend to conic optimization problems with general non-polyhedral convex cones because the conic inequalities introduce an additional nonlinear coupling between the variables.…
Convex optimization problems arising in applications often have favorable objective functions and complicated constraints, thereby precluding first-order methods from being immediately applicable. We describe an approach that exchanges the…
In this paper, we showed that for some given suitable density and pressure, there exist infinitely many compactly supported solutions with prescribed energy profile. The proof is mainly based on the convex integration scheme. We construct…
An optimization algorithm for nonsmooth nonconvex constrained optimization problems with upper-C2 objective functions is proposed and analyzed. Upper-C2 is a weakly concave property that exists in difference of convex (DC) functions and…
In this paper, we study the solution uniqueness of an individual feasible vector of a class of convex optimization problems involving convex piecewise affine functions and subject to general polyhedral constraints. This class of problems…
In this paper, Kernel PCA is reinterpreted as the solution to a convex optimization problem. Actually, there is a constrained convex problem for each principal component, so that the constraints guarantee that the principal component is…
This paper presents an algorithmic framework for the minimization of strictly convex quadratic functions. The framework is flexible and generic. At every iteration the search direction is a linear combination of the negative gradient, as…
Multicriterion optimization and Pareto optimality are fundamental tools in economics. In this paper we propose a new relaxation method for solving multiple objective quadratic programming problems. Exploiting the technique of the linear…
In this paper, a new one-parameter filled function approach is developed for nonlinear multi-objective optimization. Inspired by key filled function ideas from single-objective optimization, the proposed method is adapted to the…
We study a family of (potentially non-convex) constrained optimization problems with convex composite structure. Through a novel analysis of non-smooth geometry, we show that proximal-type algorithms applied to exact penalty formulations of…
In the paper, we develop a composite version of Mirror Prox algorithm for solving convex-concave saddle point problems and monotone variational inequalities of special structure, allowing to cover saddle point/variational analogies of what…