A New Use of Douglas-Rachford Splitting and ADMM for Identifying Infeasible, Unbounded, and Pathological Conic Programs

In this paper, we present a method for identifying infeasible, unbounded, and pathological conic programs based on Douglas-Rachford splitting, or equivalently ADMM. When an optimization program is infeasible, unbounded, or pathological, the iterates of Douglas-Rachford splitting diverge. Somewhat surprisingly, such divergent iterates still provide useful information, which our method uses for identification. In addition, for strongly infeasible problems the method produces a separating hyperplane and informs the user on how to minimally modify the given problem to achieve strong feasibility. As a first-order method, the proposed algorithm relies on simple subroutines, and therefore is simple to implement and has low per-iteration cost

Iteratively Linearized Reweighted Alternating Direction Method of Multipliers for a Class of Nonconvex Problems

In this paper, we consider solving a class of nonconvex and nonsmooth problems frequently appearing in signal processing and machine learning research. The traditional alternating direction method of multipliers encounters troubles in both mathematics and computations in solving the nonconvex and nonsmooth subproblem. In view of this, we propose a reweighted alternating direction method of multipliers. In this algorithm, all subproblems are convex and easy to solve. We also provide several guarantees for the convergence and prove that the algorithm globally converges to a critical point of an auxiliary function with the help of the Kurdyka-{\L}ojasiewicz property. Several numerical results are presented to demonstrate the efficiency of the proposed algorithm

On Positive Duality Gaps in Semidefinite Programming

We present a novel analysis of semidefinite programs (SDPs) with positive duality gaps, i.e. different optimal values in the primal and dual problems. These SDPs are extremely pathological, often unsolvable, and also serve as models of more general pathological convex programs. However, despite their allure, they are not well understood even when they have just two variables. We first completely characterize two variable SDPs with positive gaps; in particular, we transform them into a standard form that makes the positive gap trivial to recognize. The transformation is very simple, as it mostly uses elementary row operations coming from Gaussian elimination. We next show that the two variable case sheds light on larger SDPs with positive gaps: we present SDPs in any dimension in which the positive gap is caused by the same structure as in the two variable case. We analyze a fundamental parameter, the {\em singularity degree} of the duals of our SDPs, and show that it is the largest that can result in a positive gap. We finally generate a library of difficult SDPs with positive gaps (some of these SDPs have only two variables) and present a computational study.Comment: Better pictures to illustrate SDPs with positive gaps; Better version of the "double sequence" SDPs; Better literature revie

Status Determination by Interior-Point Methods for Convex Optimization Problems in Domain-Driven Form

We study the geometry of convex optimization problems given in a Domain-Driven form and categorize possible statuses of these problems using duality theory. Our duality theory for the Domain-Driven form, which accepts both conic and non-conic constraints, lets us determine and certify statuses of a problem as rigorously as the best approaches for conic formulations (which have been demonstrably very efficient in this context). We analyze the performance of an infeasible-start primal-dual algorithm for the Domain-Driven form in returning the certificates for the defined statuses. Our iteration complexity bounds for this more practical Domain-Driven form match the best ones available for conic formulations. At the end, we propose some stopping criteria for practical algorithms based on insights gained from our analyses