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.
Let
(QP) be an integer quadratic program that consists in minimizing a quadratic function subject to linear constraints. A such problem belongs to the class of
NP-Hard problems. Standard solvers that use a Branch and Bound algorithm can efficiently solve
(QP) in the specific case where its objective function is convex. Thus, to solve
(QP), we choose to reformulate it into an equivalent problem with a convex objective function. Two reformulations are possible: either we reformulate
(QP) into a linear program, or we reformulate it into a convex quadratic program. \par In the first part of this dissertation, we present several linearizations of
(QP), i.e. reformulations into an equivalent program with a linear objective function. Many linearization methods for the quadratic binary programs are known. A natural approach when considering
(QP) is therefore to reformulate it into a quadratic binary program. This can be done by the binary decomposition of each integer variable and then the linearization of each new product of two binary variables. However, this method, that we denote by \texttt{BBL (Binary Binary Linearization)}, leads to a linear program with a large number of variables and constraints. We then present a new approach, \texttt{BIL (Binary Integer Linearization)}, that consists in reformulating
(QP) into a particular quadratic integer program where each quadratic term is the product of an integer variable by a binary variable. As in the \texttt{BBL} approach, the binary variables come from the binary decomposition of the initial integer variables. Then, we linearize the obtained program by replacing each quadratic term by a new real variable and a set of inequalities. As the number of quadratic terms is lower than in the \texttt{BBL} approach, the number of additional variables is reduced. Hence, the obtained integer linear program is significantly smaller in the \texttt{BIL} approach. Moreover, contrary to what one might think, the \texttt{BIL} approach provides a better bound obtained by continuous relaxation than the \texttt{BBL} approach. Each reformulation leads to an integer linear program that is equivalent to
(QP) and that we improve by adding valid inequalities. \par In a second part, we present several quadratic convex reformulations of
(QP), i.e. we reformulate
(QP) into an equivalent program, with a quadratic convex objective function. We first introduce a simple approach to convexify
(QP) that consists in expressing linearly the squares of integer variables using their unary decompositions, and then to convexify with the smallest eigenvalue of the Hessian matrix of
(QP). We call this approach \texttt{NC (Naive Convexification)}. Then, we introduce a new convex reformulation scheme that perturbs the objective function of
(QP) with the linear expression of the products of integer variables, and the equality constraints of
(QP). Then, we show that we can compute, within this scheme, an optimal reformulation of
(QP) in terms of bound obtained by continuous relaxation : the \texttt{IQCR (Integer Quadratic Convex Reformulation)} approach. This reformulation is based on the optimal dual solution of a semi-definite relaxation of
(QP). Moreover, we show that the method \texttt{IQCR} is easily adaptable to mixed-integer programming. This adaptation, that we call \texttt{IQCRs}, also allows us to integrate the inequality constraints of
(QP) into the perturbation of the objective function of our convex reformulation scheme. Then, we present an interesting restriction of the method \texttt{IQCR}, called \texttt{CQCR (Compact Quadratic Convex Reformulation)}. The difference between this last approach and \texttt{IQCR} is that \texttt{CQCR} only uses the linear expression of the integer variable squares to perturb the objective function, while \texttt{IQCR} uses all the products. The interest is that \texttt{CQCR} produces a reformulated problem with a reduced size in comparison with the \texttt{IQCR} approach, what could be profitable experimentally. Finally, we apply these
3 methods \texttt{NC}, \texttt{CQCR} and \texttt{IQCR} to binary quadratic programming. We show that \texttt{NC} and \texttt{CQCR} are equivalent to existing binary convex reformulations. An interesting result is that \texttt{IQCR} is an improvement of existing convexifications for binary quadratic programming.\par In a third part, we design a specific Branch and Bound algorithm based on a projection property of the \texttt{IQCR} approach.\par Finally, we compare the four obtained linearizations and the three obtained quadratic convex reformulations from the computational point of view. For integer programming, computational experiences are carried out with two classes of instances of
(QP), the first having one equality constraint, and the other having one inequality constraint. The results show that most of the considered instances with up to
40 variables can be solved in one hour of CPU time by the \texttt{IQCR} and \texttt{CQCR} approaches. We then test \texttt{IQCR} on binary quadratic programming. The results corroborate that our approach \texttt{IQCR} improves existing convexifications.
We consider binary quadratic programs (QP) having a quadratic objectivefunction, linear constraints, and binary variables. Many classical solution methods ofthese problems are based on exact reformulation of QP into anequivalent mixed integer linear program. Several linearization methods werestudied in the literature. More recent solution methods alsobuild an exact reformulation but into a problem which objective function isquadratic and convex. A common point of the two approaches is that thecontinuous relaxation of the reformulated problem is a convex optimizationproblem that can be solved in polynomial time. This makes it possible to use a general branch-and-bound framework to solve the reformulated problem and even to rely on the strongness of standard solvers. In this paper, we show that several quadratic convex reformulation methods, as well as classical linearization, can be viewed within a unified framework. This shows the non-surprising result that linearization is a particular quadratic convex reformulation on the one hand. On the other hand, it allows to compare these methods from a theoretical point of view.
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.
The main objective of this work is to show the application of linear and nonlinear programming methods to deal with problems whose natural 0−1 formulation has a quadratic objective function. To this end, we develop an extension of the t-linearization technique and show how to apply it to quadratic problems such as Maximum Cut, Maximum Diversity and Maximum Edge-Weighted Clique. In addition, we review the hyperbolic smoothing technique and show how to apply it to the maximum cut problem. We carry out a theoretical study of each problem, where we particularly present the development of new valid inequalities for each of them. We perform computational experiments to evaluate the performance of each technique separately, as well as combined with the proposed new constraints. The experiments attest the quality of the bounds obtained as well as the strength of the new inequalities. We develop an exact method for each problem, based on the presented techniques. For the maximum cut and maximum edge-weighted clique, we compare our method with linearized formulations. For maximum diversity, we compare our method with linearized formulations and with our implementation of the best exact method, which does not use mathematical solvers, known in the literature. Computational experiments with instances of the literature show superior performance of our method.
Binary Quadratic Programs (BQPs) are a class of NP-hard problems that arise in a wide range of applications, including finance, machine learning, and logistics. These problems are challenging to solve due to the combinatorial search space and nonlinearity. In fact, this class of optimization problems is so challenging that, in many instances, standard algorithms struggle to identify feasible solutions within a reasonable time. Primal heuristic algorithms have been developed to quickly identify feasible solutions to BQPs. In this paper, we propose Cover-Relax-Search, an efficient primal heuristic for BQPs. This approach is inspired by multiple local search algorithms, including Undercover. We evaluate the \emph{Cover-Relax-Search} algorithm on multiple BQP benchmarks and show that our proposed heuristic identifies high-quality solutions at a faster speed and significantly reduces the primal integral compared to state-of-the-art solvers and other local search baselines.