The problem of minimizing the rank of a symmetric positive semidefinite matrix subject to constraints can be lifted to give an equivalent semidefinite program with complementarity constraints. The formulation requires two positive semidefinite matrices to be complementary. This is a continuous and nonconvex reformulation of the rank minimization problem. We develop two relaxations and show that constraint qualification holds at any stationary point of either relaxation of the rank minimization problem, and we explore the structure of the local minimizers.

中文翻译：

---

秩最小化问题的两种松弛方法

最小化受约束的对称正半定矩阵的秩的问题可以被提升以给出具有互补约束的等效半定规划。该公式要求两个半正定矩阵互补。这是秩最小化问题的连续非凸重构。我们开发了两个松弛并表明约束条件在秩最小化问题的任一松弛的任何驻点上都成立，并且我们探索了局部最小化器的结构。