A proximal gradient descent method for the extended second-order cone linear complementarity problem

Shaohua Pan, Jein Shan Chen

Research output: Contribution to journalArticle

7 Citations (Scopus)


We consider an extended second-order cone linear complementarity problem (SOCLCP), including the generalized SOCLCP, the horizontal SOCLCP, the vertical SOCLCP, and the mixed SOCLCP as special cases. In this paper, we present some simple second-order cone constrained and unconstrained reformulation problems, and under mild conditions prove the equivalence between the stationary points of these optimization problems and the solutions of the extended SOCLCP. Particularly, we develop a proximal gradient descent method for solving the second-order cone constrained problems. This method is very simple and at each iteration makes only one Euclidean projection onto second-order cones. We establish global convergence and, under a local Lipschitzian error bound assumption, linear rate of convergence. Numerical comparisons are made with the limited-memory BFGS method for the unconstrained reformulations, which verify the effectiveness of the proposed method.

Original languageEnglish
Pages (from-to)164-180
Number of pages17
JournalJournal of Mathematical Analysis and Applications
Issue number1
Publication statusPublished - 2010 Jun 1



  • Descent
  • Extended second-order cone linear complementarity problems
  • Linear convergence rate
  • Optimization reformulations
  • Proximal gradient method

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

Cite this