Compact Disjunctive Approximations to Nonconvex Quadratically Constrained Programs

Abstract

Decades of advances in mixed-integer linear programming (MILP) and recent development in mixed-integer second-order-cone programming (MISOCP) have translated very mildly to progresses in global solving nonconvex mixed-integer quadratically constrained programs (MIQCP). In this paper we propose a new approach, namely Compact Disjunctive Approximation (CDA), to approximate nonconvex MIQCP to arbitrary precision by convex MIQCPs, which can be solved by MISOCP solvers. For nonconvex MIQCP with n variables and m general quadratic constraints, our method yields relaxations with at most O(n log(1/ε)) number of continuous/binary variables and linear constraints, together with m convex quadratic constraints, where ε is the approximation accuracy. The main novelty of our method lies in a very compact lifted mixed-integer formulation for approximating the (scalar) square function. This is derived by first embedding the square function into the boundary of a three-dimensional second-order cone, and then exploiting rotational symmetry in a similar way as in the construction of BenTal-Nemirovski approximation. We further show that this lifted formulation characterize the union of finite number of simple convex sets, which naturally relax the square function in a piecewise manner with properly placed knots. We implement (with JuMP) a simple adaptive refinement algorithm. Numerical experiments on synthetic instances used in the literature show that our prototypical implementation (with hundreds of lines of Julia code) can already close a significant portion of gap left by various state-of-the-art global solvers on more difficult instances, indicating strong promises of our proposed approach.

Mixed-Projection Conic Optimization: A New Paradigm for Modeling Rank Constraints

“…A challenging predicament in separating an optimal solution to Problem (9) from ( 6)'s feasible region is that branching on the eigenvalues of Y directly-which would be the most natural extension of the branching scheme in binary optimization-is not, to our knowledge, possible. To avoid this predicament, we adapt the lifted approach proposed by Saxena et al (2010) for general mixed-integer QCQO (see also Dong and Luo 2018) to our lifted formulation (10).…”