The primal-dual hybrid gradient algorithm (PDHG), which is indeed the Arrow-Hurwicz method, has been widely used in image processing areas. However, the convergence of PDHG was established only under some restrictive conditions in the literature, and it is still missing for the case without extra constraints. In this paper, from a perspective of the variational … Read more
This short paper describes a simple subgradient-based techniques for deriving bounds on the optimal solution value when using the ADMM to solve convex optimization problems. The technique requires a bound on the magnitude of some optimal solution vector, but is otherwise completely general. Some computational examples using LASSO problems demonstrate that the technique can produce … Read more
We provide a complete classification of the extreme rays of the $6 \times 6$ copositive cone ${\cal COP}^6$. We proceed via a coarse intermediate classification of the possible minimal zero support set of an exceptional extremal matrix $A \in {\cal COP}^6$. To each such minimal zero support set we construct a stratified semi-algebraic manifold in … Read more
We consider the problem of estimating the true values of a Wiener process given noisy observations corrupted by outliers. The problem considered is closely related to the Trimmed Least Squares estimation problem, a robust estimation procedure well-studied from a statistical standpoint but poorly understood from an optimization perspective. In this paper we show how to … Read more
We consider a nonsmooth optimization problem on Riemannian manifold, whose objective function is the sum of a differentiable component and a nonsmooth convex function. We propose a manifold inexact augmented Lagrangian method (MIALM) for the considered problem. The problem is reformulated to a separable form. By utilizing the Moreau envelope, we get a smoothing subproblem … Read more
Quadratically constrained quadratic programs (QCQPs) are a fundamental class of optimization problems well-known to be NP-hard in general. In this paper we study conditions under which the standard semidefinite program (SDP) relaxation of a QCQP is tight. We begin by outlining a general framework for proving such sufficient conditions. Then using this framework, we show … Read more
In this paper, we consider multi-stage robust convex optimization problems of the minimax type. We assume that the total uncertainty set is the cartesian product of stagewise compact uncertainty sets and approximate the given problem by a sampled subproblem. Instead of looking for the worst case among the infinite and typically uncountable set of uncertain … Read more
In this paper we investigate the dual of a Multistage Stochastic Linear Program (MSLP) to study two related questions for this class of problems. The first of these questions is the study of the optimal value of the problem as a function of the involved parameters. For this sensitivity analysis problem, we provide formulas for … Read more
A stochastic second-order trust region method is proposed, which can be viewed as a second-order extension of the trust-region-ish (TRish) algorithm proposed by Curtis et al. [INFORMS J. Optim. 1(3) 200–220, 2019]. In each iteration, a search direction is computed by (approximately) solving a trust region subproblem defined by stochastic gradient and Hessian estimates. The … Read more
Polynomial optimization problems over binary variables can be expressed as integer programs using a linearization with extra monomials in addition to those arising in the given polynomial. We characterize when such a linearization yields an integral relaxation polytope, generalizing work by Del Pia and Khajavirad (SIAM Journal on Optimization, 2018) and Buchheim, Crama and Rodríguez-Heck … Read more