We consider the global optimization of nonconvex mixed-integer quadratic programs with linear equality constraints. In particular, we present a new class of convex quadratic relaxations which are derived via quadratic cuts. To construct these quadratic cuts, we solve a separation problem involving a linear matrix inequality with a special structure that allows the use of specialized solution algorithms. Our quadratic cuts are nonconvex, but define a convex feasible set when intersected with the equality constraints. We show that our relaxations are an outer-approximation of a semi-infinite convex program which under certain conditions is equivalent to a well-known semidefinite program relaxation. The new relaxations are implemented in the global optimization solver BARON, and tested by conducting numerical experiments on a large collection of problems. Results demonstrate that, for our test problems, these relaxations lead to a significant improvement in the performance of BARON.

我们考虑具有线性等式约束的非凸混合整数二次规划的全局优化。特别是，我们提出了一类新的凸二次松弛，它是通过二次切割导出的。为了构造这些二次切割，我们解决了一个涉及线性矩阵不等式的分离问题，该问题具有允许使用专门的求解算法的特殊结构。我们的二次切割是非凸的，但在与等式约束相交时定义了一个凸可行集。我们表明，我们的松弛是半无限凸程序的外近似，在某些条件下相当于众所周知的半定程序松弛。新的松弛在全局优化求解器 BARON 中实现，并通过对大量问题进行数值实验进行测试。结果表明，对于我们的测试问题，这些松弛导致 BARON 性能的显着提高。