Kronecker Product Constraints with an Application to the Two-Trust-Region Subproblem
研究了包含两个半正定约束的优化问题，通过克罗内克积构造新约束，推广了RLT技术，并用于加强双信任域子问题的半定规划松弛。
We consider semidefinite optimization problems that include constraints of the form $G(x)\succeq 0$ and $H(x)\succeq 0$, where the components of the symmetric matrices $G(\cdot)$ and $H(\cdot)$ are affine functions of $x\in\mathbb{R}^n$. In such a case we obtain a new constraint $K(x,X)\succeq 0$, where the components of $K(\cdot,\cdot)$ are affine functions of $x$ and $X$, and $X$ is an $n\times n$ matrix that is a relaxation of $xx^T$. The constraint $K(x,X)\succeq 0$ is based on the fact that $G(x)\otimes H(x)\succeq 0$, where $\otimes$ denotes the Kronecker product. This construction of a constraint based on the Kronecker product generalizes the construction of a reformation-linearization technique (RLT) constraint from two linear inequality constraints, and also the construction of a second-order cone--RLT constraint from one linear inequality constraint and a second-order cone constraint. We show how the Kronecker product constraint obtained from two second-order cone constraints can be efficiently used to computationally strengthen the semidefinite programming relaxation of the two-trust-region subproblem.