Equivalent Sufficient Conditions for Global Optimality of Quadratically Constrained Quadratic Program

\(\) We study the equivalence of several well-known sufficient optimality conditions for a general quadratically constrained quadratic program (QCQP). The conditions are classified in two categories. The first one is for determining an optimal solution and the second one is for finding an optimal value. The first category of conditions includes the existence of a … Read more

Learning to Accelerate the Global Optimization of Quadratically-Constrained Quadratic Programs

We learn optimal instance-specific heuristics for the global minimization of nonconvex quadratically-constrained quadratic programs (QCQPs). Specifically, we consider partitioning-based mixed-integer programming relaxations for nonconvex QCQPs and propose the novel problem of strong partitioning to optimally partition variable domains without sacrificing global optimality. We design a local optimization method for solving this challenging max-min strong partitioning … Read more

A Geometric View of SDP Exactness in QCQPs and its Applications

Let S denote a subset of Rn defined by quadratic equality and inequality constraints and let S denote its projected semidefinite program (SDP) relaxation. For example, take S and S to be the epigraph of a quadratically constrained quadratic program (QCQP) and the projected epigraph of its SDP relaxation respectively. In this paper, we suggest … Read more

On the tightness of SDP relaxations of QCQPs

Quadratically constrained quadratic programs (QCQPs) are a fundamental class of optimization problems well-known to be NP-hard in general. In this paper we study conditions under which the standard semidefinite program (SDP) relaxation of a QCQP is tight. We begin by outlining a general framework for proving such sufficient conditions. Then using this framework, we show … Read more

A Revisit to Quadratic Programming with One Inequality Quadratic Constraint via Matrix Pencil

The quadratic programming over one inequality quadratic constraint (QP1QC) is a very special case of quadratically constrained quadratic programming (QCQP) and attracted much attention since early 1990’s. It is now understood that, under the primal Slater condition, (QP1QC) has a tight SDP relaxation (PSDP). The optimal solution to (QP1QC), if exists, can be obtained by … Read more