VECTOR-VALUED IMPLICIT LAGRANGIAN FOR SYMMETRIC CONE COMPLEMENTARITY PROBLEMS

Lingchen Kong (), Levent Tunçel () and Naihua Xiu ()
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Lingchen Kong: Department of Applied Mathematics, Beijing Jiaotong University, Beijing 100044, P. R. China;
Levent Tunçel: Department of Combinatorics and Optimization, Faculty of Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
Naihua Xiu: Department of Applied Mathematics, Beijing Jiaotong University, Beijing 100044, P. R. China

Asia-Pacific Journal of Operational Research (APJOR), 2009, vol. 26, issue 02, 199-233

Abstract: The implicit Lagrangian was first proposed by Mangasarian and Solodov as a smooth merit function for the nonnegative orthant complementarity problem. It has attracted much attention in the past ten years because of its utility in reformulating complementarity problems as unconstrained minimization problems. In this paper, exploiting the Jordan-algebraic structure, we extend it to the vector-valued implicit Lagrangian for symmetric cone complementary problem (SCCP), and show that it is a continuously differentiable complementarity function for SCCP and whose Jacobian is strongly semismooth. As an application, we develop the real-valued implicit Lagrangian and the corresponding smooth merit function for SCCP, and give a necessary and sufficient condition for the stationary point of the merit function to be a solution of SCCP. Finally, we show that this merit function can provide a global error bound for SCCP with the uniform Cartesian P-property.

Keywords: Symmetric cone complementary problem; Jordan algebra; vector-valued implicit Lagrangian; c-function; uniform Cartesian P-property (search for similar items in EconPapers)
Date: 2009
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Citations: View citations in EconPapers (2)

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DOI: 10.1142/S0217595909002171

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