Constraint Nondegeneracy, Strong Regularity, and Nonsingularity in Semidefinite Programming
Constraint Nondegeneracy, Strong Regularity, and Nonsingularity in Semidefinite Programming
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DOI：
10.1137/070681235
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发表时间：
2008-02
期刊：
影响因子：
--
通讯作者：
Z. X. Chan;Defeng Sun
中科院分区：
文献类型：
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作者：
Z. X. Chan;Defeng Sun
已知半定规划的Karush-Kuhn-Tucker（KKT）条件可以通过对称半正定矩阵锥上的度量投影转化为非光滑系统。本文证明了该非光滑系统的原、对偶约束非退化性、强正则性、B-次微分的非奇异性以及相应的Clarke广义Jacobian在KKT点的非奇异性都是等价的。此外，我们证明了这些条件和克拉克的广义雅可比矩阵的非奇异性之间的等价性的光滑对应的这个非光滑系统中使用的几个全局收敛的光滑牛顿方法。特别地，我们建立了这些方法的原始和对偶约束下的非退化，但没有严格互补的二次收敛。
It is known that the Karush-Kuhn-Tucker (KKT) conditions of semidefinite programming can be reformulated as a nonsmooth system via the metric projector over the cone of symmetric and positive semidefinite matrices. We show in this paper that the primal and dual constraint nondegeneracies, the strong regularity, the nonsingularity of the B-subdifferential of this nonsmooth system, and the nonsingularity of the corresponding Clarke's generalized Jacobian, at a KKT point, are all equivalent. Moreover, we prove the equivalence between each of these conditions and the nonsingularity of Clarke's generalized Jacobian of the smoothed counterpart of this nonsmooth system used in several globally convergent smoothing Newton methods. In particular, we establish the quadratic convergence of these methods under the primal and dual constraint nondegeneracies, but without the strict complementarity.