Inner Approximations of Completely Positive Reformulations of Mixed Binary Quadratic Optimization Problems: A Unified Analysis∗

August 24, 2015 Abstract Every quadratic optimization problem with a mix of continuous and binary variables can be equivalently reformulated as a completely positive optimization problem, i.e., a linear optimization problem over the convex but computationally intractable cone of completely positive matrices. In this paper, we focus on general inner approximations of the cone of completely positive matrices on instances of completely positive optimization problems that arise from the reformulation of mixed binary quadratic optimization problems. We provide a characterization of the feasibility of such an inner approximation as well as the optimal value of a feasible inner approximation. For polyhedral inner approximations, our characterization implies that computing the corresponding approximate solution reduces to an optimization problem over a finite set. Our characterization yields, as a byproduct, an upper bound on the gap between the optimal value of an inner approximation and that of the original instance. We discuss the implications of the error bound for standard and box-constrained quadratic optimization problems. ∗This work was supported in part by TÜBİTAK (The Scientific and Technological Research Council of Turkey) Grant 112M870. †Department of Industrial Engineering, Koç University, 34450 Sarıyer, Istanbul, Turkey ([email protected]). The author was supported in part by TÜBA-GEBİP (Turkish Academy of Sciences Young Scientists Award Program).