A penalty method for rank minimization problems in symmetric matrices
A penalty method for rank minimization problems in symmetric matrices
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DOI：
10.1007/s10589-018-0010-6
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发表时间：
2017-01
影响因子：
2.2
通讯作者：
Xin Shen-;J. Mitchell
中科院分区：
文献类型：
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作者：
Xin Shen-;J. Mitchell
有约束的对称正半定矩阵的秩最小化问题可以等价地转化为具有互补约束的半定规划问题。该公式要求两个正半定矩阵互为补。这是秩最小化问题的连续非凸重新表述。我们研究了SDCMPCC公式的局部最优解的镇定性，从而证明了任何局部最优解都是一个KKT点。我们提出了这个问题的惩罚形式。我们给出了惩罚公式的局部最优解的镇定性结果。我们还开发了一种惩罚公式的近端交替线性化最小化（PALM）方案，并研究了将动量项纳入算法。给出了计算结果。
The problem of minimizing the rank of a symmetric positive semidefinite matrix subject to constraints can be cast equivalently as a semidefinite program with complementarity constraints (SDCMPCC). The formulation requires two positive semidefinite matrices to be complementary. This is a continuous and nonconvex reformulation of the rank minimization problem. We investigate calmness of locally optimal solutions to the SDCMPCC formulation and hence show that any locally optimal solution is a KKT point. We develop a penalty formulation of the problem. We present calmness results for locally optimal solutions to the penalty formulation. We also develop a proximal alternating linearized minimization (PALM) scheme for the penalty formulation, and investigate the incorporation of a momentum term into the algorithm. Computational results are presented.