Vector-Valued Implicit Lagrangian for Symmetric Cone complementarity Problems
Vector-Valued Implicit Lagrangian for Symmetric Cone complementarity Problems
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DOI：
10.1142/s0217595909002171
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发表时间：
2009-04
期刊：
影响因子：
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通讯作者：
Lingchen Kong;L. Tunçel;N. Xiu
中科院分区：
文献类型：
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作者：
Lingchen Kong;L. Tunçel;N. Xiu
隐式拉格朗日量首先由Mangasarian和Solodov提出，作为非负正交互补问题的光滑价值函数。由于它在将互补问题转化为无约束极小化问题中的实用性，在过去的十年中引起了人们的广泛关注。本文利用Jordan代数结构，将其推广到对称锥互补问题（SCCP）的向量值隐式Lagrange函数，并证明了它是SCCP的连续可微互补函数，其Jacobian是强半光滑的.作为应用，我们给出了SCCP的实值隐式拉格朗日量和相应的光滑价值函数，并给出了价值函数的驻点为SCCP解的充要条件.最后，我们证明了这个价值函数可以提供一个全局误差界的SCCP与一致的笛卡尔P-性质。
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.