Stability analysis of variational inequalities and nonlinear complementarity problems, via the mixed linear complementarity problem and degree theory
Stability analysis of variational inequalities and nonlinear complementarity problems, via the mixed linear complementarity problem and degree theory
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DOI：
10.1287/moor.19.4.831
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发表时间：
1994-11
影响因子：
1.7
通讯作者：
M. Gowda;J. Pang
中科院分区：
文献类型：
--
作者：
M. Gowda;J. Pang
研究了混合线性互补问题及其变形在非线性互补问题和变分不等式Karush-Kuhn-Tucker系统稳定性分析中的作用.在非奇异性假设下，混合线性互补问题可以转化为标准问题，后者的丰富理论可以直接应用于前者。在这篇文章中，我们利用度理论得到了混合线性互补问题解存在的一些充分条件。其次，我们将此存在性理论推广到混合非线性互补问题，并在一定的度理论假设下，建立了线性化问题的主要稳定性结果。然后，我们专门这个稳定性结果及其后果的参数变分不等式问题的假设下，一组独特的乘数。最后，我们考虑了后一个问题，用凸性假设代替了乘子的唯一性假设，并在一些弱的二阶条件下得到了稳定性结果。除了混合线性互补问题的新的存在性结果外，本文在稳定性范畴的主要贡献如下：解决了一个关于参数变分不等式局部可解性的猜想，利用广义线性互补问题作为工具拓宽了二阶条件，在一些弱假设下，利用解的孤立性刻画了线性互补问题和仿射变分不等式问题解的稳定性，以及在一些弱二阶条件下的各种稳定性定理.
This paper is concerned with the mixed linear complementarity problem and the role it and its variants play in the stability analysis of the nonlinear complementarity problem and the Karush-Kuhn-Tucker system of a variational inequality problem. Under a nonsingular assumption, the mixed linear complementarity problem can be converted to the standard problem; in this case, the rich theory of the latter can be directly applied to the former. In this work, we employ degree theory to derive some sufficient conditions for the existence of a solution to the mixed linear complementarity problem in the absence of the nonsingularity property. Next, we extend this existence theory to the mixed nonlinear complementarity problem and establish a main stability result under a certain degree-theoretic assumption concerning the linearized problem. We then specialize this stability result and its consequences to the parametric variational inequality problem under the assumption of a unique set of multipliers. Finally, we consider the latter problem with the uniqueness assumption of the multipliers replaced by a convexity assumption and obtain stability results under some weak second-order conditions. In addition to the new existence results for the mixed linear complementarity problem, the main contributions of this paper in the stability category are the following: a resolution to a conjecture concerning the local solvability of a parametric variational inequality, the use of the generalized linear complementarity problem as a tool to broaden the second-order conditions, the characterization of the solution stability of the linear complementarity problem and the affine variational inequality problem in terms of the solution isolatedness under some weak hypotheses, and various stability theorems under some weak second-order conditions.