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Y. Gorkem Gokmen (gorkemgokmengmail.com) Abstract: 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 socalled 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 socalled 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. Keywords: Standard quadratic programs, copositive cone, completely positive cone, doubly nonnegative relaxation Category 1: Nonlinear Optimization (Quadratic Programming ) Category 2: Linear, Cone and Semidefinite Programming Citation: ERGO Technical Report No. 20001, The University of Edinburgh, Edinburgh, UK Download: [PDF] Entry Submitted: 02/21/2020 Modify/Update this entry  
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