We study convex relaxations of nonconvex quadratic programs. We identify a family of so-called feasibility preserving convex relaxations, which includes the well-known copositive and doubly nonnegative relaxations, with the property that the convex relaxation is feasible if and only if the nonconvex quadratic program is feasible. We observe that each convex relaxation in this family implicitly induces a convex underestimator of the objective function on the feasible region of the quadratic program. This alternative perspective on convex relaxations enables us to establish several useful properties of the corresponding convex underestimators. In particular, if the recession cone of the feasible region of the quadratic program does not contain any directions of negative curvature, we show that the convex underestimator arising from the copositive relaxation is precisely the convex envelope of the objective function of the quadratic program, strengthening Burer’s well-known result on the exactness of the copositive relaxation in the case of nonconvex quadratic programs. We also present an algorithmic recipe for constructing instances of quadratic programs with a finite optimal value but an unbounded relaxation for a rather large family of convex relaxations including the doubly nonnegative relaxation.

我们研究非凸二次程序的凸松弛。我们确定了一系列所谓的保持可行性的凸松弛，其中包括众所周知的共正松弛和双非负松弛，具有以下特性：当且仅当非凸二次规划可行时，凸松弛才是可行的。我们观察到，该族中的每个凸松弛都隐含地诱导了二次规划可行区域上目标函数的凸低估量。这种关于凸松弛的替代观点使我们能够建立相应凸低估量的几个有用属性。特别地，如果二次规划可行区域的衰退锥不包含任何负曲率方向，我们表明，由共正松弛产生的凸低估量恰好是二次规划目标函数的凸包络，加强了 Burer 在非凸二次规划情况下关于共正松弛的精确性的众所周知的结果。我们还提出了一种算法配方，用于构造具有有限最优值但对于包括双非负松弛在内的相当大的凸松弛族的无限松弛的二次程序实例。