The problem of minimizing a (nonconvex) quadratic form over the unit simplex, referred to as a standard quadratic program, admits an exact convex conic formulation over the computationally intractable cone of completely positive matrices. Replacing the intractable cone in this formulation by the larger but tractable cone of doubly nonnegative matrices, i.e., the cone of positive semidefinite and componentwise nonnegative matrices, one obtains the so-called doubly nonnegative relaxation, whose optimal value yields a lower bound on that of the original problem. We present a full algebraic characterization of the set of instances of standard quadratic programs that admit an exact doubly nonnegative relaxation. This characterization yields an algorithmic recipe for constructing such an instance. In addition, we explicitly identify three families of instances for which the doubly nonnegative relaxation is exact. We establish several relations between the so-called convexity graph of an instance and the tightness of the doubly nonnegative relaxation. We also provide an algebraic characterization of the set of instances for which the doubly nonnegative relaxation has a positive gap and show how to construct such an instance using this characterization.

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关于具有精确和非精确双重非负松弛的标准二次规划

在单位单纯形上最小化（非凸）二次形式的问题，称为标准二次规划，允许在完全正矩阵的计算上难以处理的锥体上使用精确的凸圆锥公式。用更大但易处理的双非负矩阵圆锥代替该公式中的难处理锥，即正半定和分量非负矩阵的圆锥，可以获得所谓的双非负松弛，其最优值产生下界原来的问题。我们提出了标准二次程序的实例集的完整代数特征，这些实例允许精确的双非负松弛。这种表征产生了构建这样一个实例的算法配方。此外，我们明确地确定了双非负松弛是精确的三个实例族。我们在实例的所谓凸图和双非负松弛的紧密度之间建立了几种关系。我们还提供了双非负松弛具有正间隙的实例集的代数表征，并展示了如何使用此表征构建这样的实例。