Abstract. We develop a spatial branch-and-cut approach for nonconvex Quadratically Constrained Quadratic Programs with bounded complex variables (CQCQP). Linear valid inequalities are added at each node of the search tree to strengthen semidefinite programming relaxations of CQCQP. These valid inequalities are derived from the convex hull description of a nonconvex set of 2 × 2 positive semidefinite Hermitian matrices subject to a rank-one constraint. We propose branching rules based on an alternative to the rank-one constraint that allows for local measurement of constraint violation. Closed-form bound tightening procedures are used to reduce the domain of the problem. We apply the algorithm to solve the Alternating Current Optimal Power Flow problem with complex variables as well as the Box-constrained Quadratic Programming problem with real variables.
Cutting planes are derived from specific problem structures, such as a single linear constraint from an integer program. This paper introduces cuts that involve minimal structural assumptions, enabling the generation of strong polyhedral relaxations for a broad class of problems. We consider valid inequalities for the set S ∩ P , where S is a closed set, and P is a polyhedron. Given an oracle that provides the distance from a point to S we construct a pure cutting plane algorithm; if the initial relaxation is a polytope, the algorithm is shown to converge. Cuts are generated from convex forbidden zones, or S-free sets derived from the oracle. We also consider the special case of polynomial optimization. Polynomial optimization may be represented using a symmetric matrix of variables, and in this lifted representation we can let S be the set of matrices that are real, symmetric outer products. Accordingly we develop a theory of outer-product-free sets. All maximal outer-product-free sets of full dimension are shown to be convex cones and we identify two families of such sets. These families can be used to generate intersection cuts that can separate any infeasible extreme point of a linear programming relaxation in polynomial time. Moreover, in the special case of polynomial optimization we derive strengthened oracle-based intersection cuts that can also ensure separation in polynomial time.arXiv:1610.04604v5 [math.OC]
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