This paper addresses the solution of a nonlinear boolean quadratic programming problem using three different approaches. The first uses a classic linearization technique to transform the original problem into a Mixed Integer Linear Programming (MILP) problem for which multiple formulations are studied. The second uses the capabilities of current MILP solvers to deal with Mixed Integer Quadratic Programming (MIQP) problems, and the third relies on the utilization of optimization solvers available that can deal with nonlinear combinatorial problems. Two additional strategies relying on the MILP formulations are evaluated. Special emphasis is placed on the definition of computationally efficient MILP reformulations and their comparison with other approaches. The results indicate that the most efficient approach relies on solving the problem as a MIQP problem using a specific MILP solver from the set of solvers tested, while the most efficient MILP formulations are still a better option than solving the problem as a MIQP using the remaining MILP solvers tested, and their performance is considerably superior to the application of general purpose solvers.
This paper addresses the solution of a nonlinear boolean quadratic programming problem using three different approaches. The first uses a classic linearization technique to transform the original problem into a Mixed Integer Linear Programming (MILP) problem for which multiple formulations are studied. The second uses the capabilities of current MILP solvers to deal with Mixed Integer Quadratic Programming (MIQP) problems, and the third relies on the utilization of optimization solvers available that can deal with nonlinear combinatorial problems. Two additional strategies relying on the MILP formulations are evaluated. Special emphasis is placed on the definition of computationally efficient MILP reformulations and their comparison with other approaches. The results indicate that the most efficient approach relies on solving the problem as a MIQP problem using a specific MILP solver from the set of solvers tested, while the most efficient MILP formulations are still a better option than solving the problem as a MIQP using the remaining MILP solvers tested, and their performance is considerably superior to the application of general purpose solvers.
We reformulate a (indefinite) quadratic program (QP) as a mixed-integer linear programming (MILP) problem by first reformulating a QP as a linear complementary problem, and then using binary variables and big-M constraints to model its complementary constraints. To obtain such reformulation, we use fundamental results on the solution of perturbed linear systems to impose bounds on the QP’s dual variables without eliminating any of its (globally) optimal primal solutions. Reformulating a nonconvex QP as a MILP problem allows the use of current state-of-the-art MILP solvers to find its global optimal solution. To illustrate this, we compare the performance of this MILP-based solution approach, labeled quadprogIP, with quadprogBB, BARON, and CPLEX. In practice, quadprogIP is shown to typically outperform by orders of magnitude quadprogBB, BARON, and CPLEX on standard QPs. Also, unlike quadprogBB, quadprogIP is able to solve QP instances in which the dual feasible set is unbounded. The MATLAB code quadprogIP and the instances used to perform the reported numerical experiments are publicly available at https://github.com/xiawei918/quadprogIP .
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.
This paper is concerned with binary quadratic programs (BQPs), which are among the most well-studied classes of nonlinear integer optimization problems because of their wide variety of applications. While a number of different solution approaches have been proposed for tackling BQPs, practitioners need techniques that are both efficient and easy to implement. We revisit two of the most widely used linearization strategies for BQPs and examine the effectiveness of enhancements to these formulations that have been suggested in the literature. We perform a detailed large-scale computational study over five different classes of BQPs to compare these two linearizations with a more recent linear reformulation and direct submission of the nonlinear integer program to an optimization solver. The goal is to provide practitioners with guidance on how to best approach solving BQPs in an effective and easily implemented manner.
In this paper, we propose a general modeling framework for a large class of binary quadratic programs subject to variable partitioning constraints.This problem has a wide range of applications as many of the binary quadratic programs with linear constraints can be represented in this form.By exploiting the problems' structure, we propose mixed-integer nonlinear program (MINLP) and mixed-integer linear program (MILP) reformulations and show the relationship between the two models in terms of the relaxation strength.Our methodology relies on a convex reformulation of the proposed MINLP and a branch-and-cut algorithm based on outer approximation cuts where the cuts are generated on the fly by efficiently solving separation subproblems.Our experimental results on various quadratic combinatorial optimization problems show that our approach outperforms the state-of-the-art solver applied to different MILP reformulations of the corresponding problems.
We consider in this paper quadratic programming problems with cardinality and minimum threshold constraints that arise naturally in various real-world applications such as portfolio selection and subset selection in regression. This class of problems can be formulated as mixed-integer 0-1 quadratic programs. We propose a new semidefinite program (SDP) approach for computing the “best” diagonal decomposition that gives the tightest continuous relaxation of the perspective reformulation of the problem. We also give an alternative way of deriving the perspective reformulation by applying a special Lagrangian decomposition scheme to the diagonal decomposition of the problem. This derivation can be viewed as a “dual” method to the convexification method employing the perspective function on semicontinuous variables. Computational results show that the proposed SDP approach can be advantageous for improving the performance of mixed-integer quadratic programming solvers when applied to the perspective reformulations of the problem.
We present an iterative algorithm to compute feasible solutions in reasonable running time to quadratically constrained quadratic programs (QCQPs), which form a challenging class of nonconvex continuous optimization. This algorithm is based on a mixed-integer linear program (MILP) which is a restriction of the original QCQP obtained by discretizing all quadratic terms. In each iteration, this MILP restriction is solved to get a feasible QCQP solution. Since the quality of this solution heavily depends on the chosen discretization of the MILP, we iteratively adapt the discretization values based on the MILP solution of the previous iteration. To maintain a reasonable problem size in each iteration of the algorithm, the discretization sizes are fixed at predefined values. Although our algorithm did not always yield good feasible solutions on arbitrary QCQP instances, an extensive computational study on almost 1300 test instances of two different problem classes - box-constrained quadratic programs with complementarity constraints and disjoint bilinear programs, demonstrates the effectiveness of our approach. We compare the quality of our solutions against those from heuristics and local optimization algorithms in two state-of-the-art commercial solvers and observe that on one instance class we clearly outperform the other methods whereas on the other class we obtain competitive results.