Interior Point Trajectories and a Homogeneous Model for Nonlinear Complementarity Problems over Symmetric Cones
Interior Point Trajectories and a Homogeneous Model for Nonlinear Complementarity Problems over Symmetric Cones
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DOI：
10.1137/04061427x
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发表时间：
2006-12
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影响因子：
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通讯作者：
Akiko Yoshise
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文献类型：
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作者：
Akiko Yoshise
我们研究了在对称锥上解决单调非线性混合互补性问题的连续轨迹。而[L。的分析Faybusovich，《阳性》，1（1997），pp。331-357]取决于凸对数栏函数的优化理论，我们的方法基于蒙特罗和pang [Math [Math]的论文。操作。 Res。，23（1998），第39-60页]，其中在对称阳性半芬矿矩阵的锥体上显示了有关连续轨迹的大量结论。作为结果的应用，我们为对称锥上标准单调非线性互补性问题提出了一个均质模型，并讨论其理论方面。因此，我们显示了具有以下特性的路径的存在：（a）路径是有界的，并且具有微不足道的起点，而没有任何规律性假设，即存在可行或严格可行的解决方案。 （b）路径的任何积累点都是均匀模型的解决方案。 （c）如果原始问题是可以解决的，则路径的每个累积点都会为我们提供有限的解决方案。 （d）如果原始问题是强烈不可行的，那么，在Lipschitz的连续性的假设下，路径的任何积累点都会使我们获得有限的证书，证明是不可行的。
We study the continuous trajectories for solving monotone nonlinear mixed complementarity problems over symmetric cones. While the analysis in [L. Faybusovich, Positivity, 1 (1997), pp. 331-357] depends on the optimization theory of convex log-barrier functions, our approach is based on the paper of Monteiro and Pang [Math. Oper. Res., 23 (1998), pp. 39-60], where a vast set of conclusions concerning continuous trajectories is shown for monotone complementarity problems over the cone of symmetric positive semidefinite matrices. As an application of the results, we propose a homogeneous model for standard monotone nonlinear complementarity problems over symmetric cones and discuss its theoretical aspects. Consequently, we show the existence of a path having the following properties: (a) The path is bounded and has a trivial starting point without any regularity assumption concerning the existence of feasible or strictly feasible solutions. (b) Any accumulation point of the path is a solution of the homogeneous model. (c) If the original problem is solvable, then every accumulation point of the path gives us a finite solution. (d) If the original problem is strongly infeasible, then, under the assumption of Lipschitz continuity, any accumulation point of the path gives us a finite certificate proving infeasibility.