Tutte colloquium-Arnesh Sujanani

Title:The Inexact Augmented Lagrangian Method: Optimal Complexity Bounds and Applications to Solving Huge SDPs

Abstract:In the first part of this talk, we present optimal and nearly-optimal parameter-free augmented Lagrangian (AL) methods for convex and strongly optimization with linear constraints. Our AL methods employ tractable inexact criteria for solving their inner subproblems, which accelerated methods can be shown to achieve in a finite number of iterations that depends on the target accuracy.

In the second part of this talk, we present a new inexact augmented Lagrangian method, namely, HALLaR, that employs a Burer-Monteiro style low-rank factorization for solving large-scale semidefinite programs (SDPs). The AL subproblems are solved by a hybrid low-rank method, which is based on a combination of a low-rank Frank-Wolfe method and a nonconvex accelerated inexact proximal point method. In contrast to the classical low-rank method by Burer and Monteiro, HALLaR finds a near-optimal solution (with provable complexity bounds) of SDP instances satisfying strong duality. Computational results comparing HALLaR to state-of-the-art solvers on several large SDP instances show that the former finds higher accurate solutions in substantially less CPU time than the latter ones. For example, in less than 20 minutes, HALLaR can solve (on a personal laptop) a maximum stable set SDP with 1 million vertices and 10 million edges within 1e-5 relative accuracy.

This talk is based on joint work with Saeed Ghadimi and Henry Wolkowicz from University of Waterloo and Diego Cifuentes and Renato Monteiro from Georgia Tech.