TY - JOUR
T1 - A proximal gradient descent method for the extended second-order cone linear complementarity problem
AU - Pan, Shaohua
AU - Chen, Jein Shan
N1 - Funding Information:
E-mail addresses: [email protected] (S. Pan), [email protected] (J.-S. Chen). 1 The author’s work is supported by National Young Natural Science Foundation (No. 10901058) and Guangdong Natural Science Foundation (No.
Funding Information:
9251802902000001). 2 Member of Mathematics Division, National Center for Theoretical Sciences, Taipei Office. The author’s work is partially supported by National Science Council of Taiwan.
PY - 2010/6/1
Y1 - 2010/6/1
N2 - 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.
AB - 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.
KW - Descent
KW - Extended second-order cone linear complementarity problems
KW - Linear convergence rate
KW - Optimization reformulations
KW - Proximal gradient method
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U2 - 10.1016/j.jmaa.2010.01.011
DO - 10.1016/j.jmaa.2010.01.011
M3 - Article
AN - SCOPUS:75749102216
SN - 0022-247X
VL - 366
SP - 164
EP - 180
JO - Journal of Mathematical Analysis and Applications
JF - Journal of Mathematical Analysis and Applications
IS - 1
ER -