QSDPNAL: A Two-phase Augmented Lagrangian Method for Convex Quadratic Semidefinite Programming

Xudong Li, Defeng Sun, Kim Chuan Toh

Research output: Journal article publicationJournal articleAcademic researchpeer-review

33 Citations (Scopus)


In this paper, we present a two-phase augmented Lagrangian method, called QSDPNAL, for solving convex quadratic semidefinite programming (QSDP) problems with constraints consisting of a large number of linear equality and inequality constraints, a simple convex polyhedral set constraint, and a positive semidefinite cone constraint. A first order algorithm which relies on the inexact Schur complement based decomposition technique is developed in QSDPNAL-Phase I with the aim of solving a QSDP problem to moderate accuracy or using it to generate a reasonably good initial point for the second phase. In QSDPNAL-Phase II, we design an augmented Lagrangian method (ALM) wherein the inner subproblem in each iteration is solved via inexact semismooth Newton based algorithms. Simple and implementable stopping criteria are designed for the ALM. Moreover, under mild conditions, we are able to establish the rate of convergence of the proposed algorithm and prove the R-(super)linear convergence of the KKT residual. In the implementation of QSDPNAL, we also develop efficient techniques for solving large scale linear systems of equations under certain subspace constraints. More specifically, simpler and yet better conditioned linear systems are carefully designed to replace the original linear systems and novel shadow sequences are constructed to alleviate the numerical difficulties brought about by the crucial subspace constraints. Extensive numerical results for various large scale QSDPs show that our two-phase algorithm is highly efficient and robust in obtaining accurate solutions. The software reviewed as part of this submission was given the DOI (Digital Object Identifier) https://doi.org/10.5281/zenodo.1206980.

Original languageEnglish
Pages (from-to)703-743
Number of pages41
JournalMathematical Programming Computation
Issue number4
Publication statusPublished - 1 Dec 2018


  • Augmented Lagrangian
  • Inexact semismooth Newton method
  • Quadratic semidefinite programming
  • Schur complement

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Software


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