Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A low-rank augmented Lagrangian method for large-scale semidefinite programming based on a hybrid convex-nonconvex approach

Renato D. C. Monteiro, Arnesh Sujanani, Diego Cifuentes

Abstract

This paper introduces HALLaR, a new first-order method for solving large-scale semidefinite programs (SDPs) with bounded domain. HALLaR is an inexact augmented Lagrangian (AL) method where the AL subproblems are solved by a novel hybrid low-rank (HLR) method. The recipe behind HLR is based on two key ingredients: 1) an adaptive inexact proximal point method with inner acceleration; 2) Frank-Wolfe steps to escape from spurious local stationary points. In contrast to the low-rank method of 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 arising from maximum stable set, phase retrieval, and matrix completion 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 a maximum stable set SDP instance with dimension pair $(n,m)\approx (10^6,10^7)$ within $10^{-5}$ relative precision.

A low-rank augmented Lagrangian method for large-scale semidefinite programming based on a hybrid convex-nonconvex approach

Abstract

This paper introduces HALLaR, a new first-order method for solving large-scale semidefinite programs (SDPs) with bounded domain. HALLaR is an inexact augmented Lagrangian (AL) method where the AL subproblems are solved by a novel hybrid low-rank (HLR) method. The recipe behind HLR is based on two key ingredients: 1) an adaptive inexact proximal point method with inner acceleration; 2) Frank-Wolfe steps to escape from spurious local stationary points. In contrast to the low-rank method of 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 arising from maximum stable set, phase retrieval, and matrix completion 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 a maximum stable set SDP instance with dimension pair within relative precision.
Paper Structure (30 sections, 29 theorems, 126 equations, 9 tables)

This paper contains 30 sections, 29 theorems, 126 equations, 9 tables.

Table of Contents

  1. Introduction
  2. Low-rank approach.
  3. Outline of HALLaR.
  4. Computational impact.
  5. Related works.
  6. Structure of the paper.
  7. Basic Definitions and Notations
  8. Hybrid Low-Rank Method
  9. Problem of Interest and Solution Type
  10. The ADAP-AIPP Method
  11. HLR Method
  12. Proof of Theorem \ref{['thm:complexityLRHFW']}
  13. HALLaR
  14. Description of HALLaR and Main Theorem
  15. Proof of Theorem \ref{['thm:OuterComplexitySDP']}
  16. ...and 15 more sections

Key Result

Lemma 2.1

Let $Z \in \Delta^n$ be given and define Then:

Theorems & Definitions (59)

  • Lemma 2.1
  • proof
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • Theorem 2.5
  • Proposition 2.6
  • proof
  • Proposition 2.7
  • ...and 49 more