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Journal of Industrial and Management Optimization

A fast global algorithm for multi-linearly constrained separable binary quadratic program

Guangtai Qu, Shaoze Li, Zhibin Deng, Cheng Zhu Lu

Published Oct 9, 2024• 5 citations
Publisher

Abstract

The multi-linearly constrained separable binary quadratic programming problem (MLSBQP) is a direct extension of the classical singly-linearly constrained separable binary quadratic programming problem (SLSBQP). In the literature, several algorithms have been proposed for SLSBQP which can solve problem instances with millions of variables globally. However, for the case of MLSBQP, the current state-of-the-art solver can only solve problems with up to 40000 binary variables within a reasonable computational time. In this paper, we propose an extremely efficient branch-and-bound algorithm for solving MLSBQP. The main feature of the proposed algorithm is that an efficient dual coordinate ascent algorithm is applied to obtain lower bounds, and the complexity of each inner loop of the dual coordinate ascent algorithm is $ \mathcal{O}(n\log n) $. Our numerical experiments show that the proposed branch-and-bound algorithm can solve MLSBQP with up to millions of variables within only several minutes, and performs much faster than Gurobi when the number of linear constraints is small.

DOI
10.3934/jimo.2024132

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The multi-linearly constrained separable binary quadratic programming problem (MLSBQP) is a direct extension of the classical singly-linearly constrained separable binary quadratic programming problem (SLSBQP). In the literature, several algorithms have been proposed for SLSBQP which can solve problem instances with millions of variables globally. However, for the case of MLSBQP, the current state-of-the-art solver can only solve problems with up to 40000 binary variables within a reasonable computational time. In this paper, we propose an extremely efficient branch-and-bound algorithm for solving MLSBQP. The main feature of the proposed algorithm is that an efficient dual coordinate ascent algorithm is applied to obtain lower bounds, and the complexity of each inner loop of the dual coordinate ascent algorithm is O(nlog⁡n)\mathcal{O}(n\log n)O(nlogn). Our numerical experiments show that the proposed branch-and-bound algorithm can solve MLSBQP with up to millions of variables within only several minutes, and performs much faster than Gurobi when the number of linear constraints is small.
Binary programs with a quadratic objective function are NP-hard in general, even if the linear optimization problem over the same feasible set is tractable. In this paper, we address such problems by computing quadratic global underestimators of the objective function that are separable but not necessarily convex. Exploiting the binary constraint on the variables, a minimizer of the separable underestimator over the feasible set can be computed by solving an appropriate linear minimization problem over the same feasible set. Embedding the resulting lower bounds into a branch-and-bound framework, we obtain an exact algorithm for the original quadratic binary program. The main practical challenge is the fast computation of an appropriate underestimator, which in our approach reduces to solving a series of semidefinite programs. We exploit the special structure of the resulting problems to obtain a tailored coordinate-descent method for their solution. Our extensive experimental results on various quadratic combinatorial optimization problems show that our approach outperforms both CPLEX and the related QCR method as well as the SDP-based software BiqCrunch on instances of the quadratic shortest path problem and the quadratic assignment problem.
The class of mixed-integer quadratically constrained quadratic programs (QCQP) consists of minimizing a quadratic function under quadratic constraints where the variables could be integer or continuous. On a previous paper we introduced a method called MIQCR for solving QCQPs with the following restriction: all quadratic sub-functions of purely continuous variables are already convex. In this paper, we propose an extension of MIQCR which applies to any QCQP. Let (P) be a QCQP. Our approach to solve (P) is first to build an equivalent mixed-integer quadratic problem (P∗). This equivalent problem (P∗) has a quadratic convex objective function, linear constraints, and additional variables y that are meant to satisfy the additional quadratic constraints y=xxT, where x are the initial variables of problem (P). We then propose to solve (P∗) by a branch-and-bound algorithm based on the relaxation of the additional quadratic constraints and of the integrality constraints. This type of branching is known as spatial branch-and-bound. Computational experiences are carried out on a total of 325 instances. The results show that the solution time of most of the considered instances is improved by our method in comparison with the recent implementation of QuadProgBB, and with the solvers Cplex, Couenne, Scip, BARON and GloMIQO.
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.
We present BiqBin, an exact solver for linearly constrained binary quadratic problems. Our approach is based on an exact penalty method to first efficiently transform the original problem into an instance of Max-Cut, and then to solve the Max-Cut problem by a branch-and-bound algorithm. All the main ingredients are carefully developed using new semidefinite programming relaxations obtained by strengthening the existing relaxations with a set of hypermetric inequalities, applying the bundle method as the bounding routine and using new strategies for exploring the branch-and-bound tree. Furthermore, an efficient C implementation of a sequential and a parallel branch-and-bound algorithm is presented. The latter is based on a load coordinator-worker scheme using MPI for multi-node parallelization and is evaluated on a high-performance computer. The new solver is benchmarked against BiqCrunch, GUROBI, and SCIP on four families of (linearly constrained) binary quadratic problems. Numerical results demonstrate that BiqBin is a highly competitive solver. The serial version outperforms the other three solvers on the majority of the benchmark instances. We also evaluate the parallel solver and show that it has good scaling properties. The general audience can use it as an on-line service available at http://www.biqbin.eu .
The unconstrained binary quadratic programming problem (QP) is a classical non-linear problem of optimizing a quadratic objective by a suitable choice of binary decision variables. This paper proposes new Lagrangean decompositions to find bounds for QP. The methods presented treat a mixed binary linear version (LQP) of QP with constraints represented by a graph. This graph is partitioned into clusters of vertices forming a dual problem that is solved by a subgradient algorithm. The subproblems formed by the generated subgraphs are solved by CPLEX. Computational experiments consider a data set formed by several difficult instances with different features. The results show the efficiency of the proposed methods over traditional Lagrangean relaxations and other methods found in the literature.
Binary quadratic programming (BQP) is a class of combinatorial optimization problems comprising binary variables, quadratic objective functions, and linear/nonlinear constraints. This paper examines a unified framework to reformulate a BQP problem with linear constraints to a new BQP with an exponential number of variables defined on a graph. This framework relies on the concept of stars in the graph to split the quadratic costs into adjacent and nonadjacent components indicating in-star and out-of-star interactions. We exploit the star-based structure of the new reformulation to develop a decomposition-based column generation algorithm. In our computational experiments, we evaluate the performance of our methodology on different applications with different quadratic structures. The quadratic component of the problem is dealt with in the column generation master problem and its subproblem. Results indicate the superiority of the framework over one of the state-of-the-art solvers, GUROBI, when applied to various benchmark reformulations with adjacent-only or sparse quadratic cost matrices. The framework outperforms GUROBI in terms of both dual bound and computational time in almost all instances. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms—Discrete. Funding: The authors thank the Mitacs Accelerate Program for providing funding for this project. In addition, B. Rostami gratefully acknowledges the funding provided by the Canadian Natural Sciences and Engineering Research Council (NSERC) under a Discovery Grant [Grant RGPIN-2020-05395]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoc.2021.0186 .