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Exploiting Low-Rank Structure in Semidefinite Programming by Approximate Operator Splitting

Mario Souto (mariohsouto***at***gmail.com)
Joaquim Dias Garcia (joaquimdgarcia***at***gmail.com )
Alvaro Veiga (alvf***at***ele.puc-rio.br )

Abstract: In contrast with many other convex optimization classes, state-of-the-art semidefinite programming solvers are yet unable to efficiently solve large scale instances. This work aims to reduce this scalability gap by proposing a novel proximal algorithm for solving general semidefinite programming problems. The proposed methodology, based on the primal-dual hybrid gradient method, allows the presence of linear inequalities without the need of adding extra slack variables and avoids solving a linear system at each iteration. More importantly, it does simultaneously compute the dual variables associated with the linear constraints. The main contribution of this work is to achieve a substantial speedup by effectively adjusting the proposed algorithm in order to exploit the low-rank property inherent to several semidefinite programming problems. This proposed modification is the key element that allows the operator splitting method to efficiently scale to larger instances. Convergence guarantees are presented along with an intuitive interpretation of the algorithm. Additionally, an open source semidefinite programming solver, called ProxSDP, is made available and implementation details are discussed. Case studies are presented in order to evaluate the performance of the proposed methodology.

Keywords: Semidefinite Programming, Operator Splitting Methods, Inexact Fixed Point Iteration, Approximate Proximal Point, Low-Rank Matrix Approximation, Convex Optimization

Category 1: Convex and Nonsmooth Optimization (Convex Optimization )

Category 2: Linear, Cone and Semidefinite Programming (Semi-definite Programming )

Category 3: Optimization Software and Modeling Systems

Citation: Unpublished: October 11, 2018. PUC-Rio (Pontifical Catholic University of Rio de Janeiro) university, Rua Marquês de São Vicente, 225, Gávea - Rio de Janeiro, RJ - Brasil - 22451-900 Cx. Postal: 38097 - Phone: (55 21) 3527-1001

Download: [PDF]

Entry Submitted: 10/11/2018
Entry Accepted: 10/12/2018
Entry Last Modified: 12/19/2018

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