Globally Solving Nonconvex Quadratic Programs via Linear Integer Programming Techniques

Abstract

Quadratic programming (QP) is a well-studied fundamental NP-hard optimization problem which optimizes a quadratic objective over a set of linear constraints. In this paper, we reformulate QPs as a mixed-integer linear problem (MILP). This is done via the reformulation of QP as a linear complementary problem, and the use of binary variables and big-M constraints, to model the complementary constraints. To obtain such reformulation, we show how to impose bounds on the dual variables without eliminating all the (globally) optimal primal solutions; using some fundamental results on the solution of perturbed linear systems.Reformulating non-convex QPs as MILPs provides an advantageous way to obtain global solutions as it allows the use of current state-of-the-art MILP solvers. To illustrate this, we compare the performance of our solution approach, labeled quadprogIP, with the current benchmark global QP solver quadprogBB, as well as with BARON, one of the leading non-linear programming (NLP) solvers, and CPLEX's non-convex QP solver, on a large variety of QP test instances. In practice, quadprogIP is shown to typically outperform by orders of magnitude quadprogBB, BARON, and CPLEX on standard QPs. For general QPs, quadprogIP outperforms quadprogBB, outperforms BARON in most instances, while CPLEX performs the best on these instances. For box-constrained QPs, quadprogIP has a comparable performance to quadprogBB and BARON in small-to medium-scale instances, but is outperformed by these solvers on large-scale instances; while CPLEX performs the best on box-constrained QP instances. Also, unlike quadprogBB, the solution approach proposed here is able to solve QP instances whose dual feasible set is unbounded. The MATLAB code, called quadprogIP, and the instances used to perform these numerical experiments are publicly available at https://github.com/xiawei918/quadprogIP. * many others).In this paper, we reformulate QPs as a mixed-integer linear problem (MILP). This provides an advantageous way to obtain global solutions as it allows the use of current state-of-the-art MILP solvers. Moreover, the numerical experiments of Section 3 show that a basic implementation of the proposed algorithm, which we refer to as quadprogIP, typically outperform by orders of magnitude quadprogBB, BARON, and CPLEX on standard QPs. For general QPs, quadprogIP outperforms quadprogBB, outperforms BARON in most instances, while CPLEX performs the best on these instances. For box-constrained QPs, quadprogIP has a comparable performance to quadprogBB and BARON in small-to medium-scale instances, but is outperformed by these solvers on large-scale instances; while CPLEX performs the best on boxconstrained QP instances.Unlike quadprogBB, the solution approach proposed here is able to solve QP instances whose dual feasible set is unbounded. The MATLAB code and the instances used to perform these numerical experiments are available at https://github.com/xiawei918/quadprogIP.To obtain the proposed MILP-reformulation (see Sec. 2.1), the QP's KKT conditi...

Multi-Tone Signal Optimization for Wireless Power Transfer in the Presence of Wireless Communication Links

“…As shown in [29], [31], the KKT conditions (20)- (22) can be used to linearize the objective of (24) as follows,…”

“…A solution method to this problem was proposed in [31] by solving the following equivalent MILP: max…”

Global solutions of nonconvex standard quadratic programs via mixed integer linear programming reformulations

“…Our work is related to the previous work on reformulations of a quadratic program as an instance of a linear program with complementarity constraints (LPCC) [20,31]. For a general quadratic program, the paper [20] proposes a two-stage LPCC approach.…”

“…An LPCC can also be formulated as a mixed integer linear programming (MILP) problem and can be solved using Benders decomposition [20] or by branch-and-cut [32]. A similar MILP formulation is proposed in [31] under the assumption of a bounded feasible region. Alternatively, using the completely positive reformulation of (StQP) (see, e.g., [7]), adaptive inner and outer polyhedral approximations of completely positive programs can be employed [11].…”

“…The resulting complementarity problems are formulated as MILP problems and solved using a parameter-free approach via Benders decomposition, which eliminates the need for big-M parameters [19] (see [32] for a branch-and-cut approach). In contrast, the paper [31] explicitly uses big-M parameters. By using a Hoffman type error bound, the authors show that there exists a valid upper bound for the big-M parameters under the assumption of a bounded feasible region.…”