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Résolution de programmes quadratiques en nombres entiers

Amélie Lambert

Published Nov 26, 2009• 1 citation
OpenAlex

Abstract

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 $\mathcal{NP}\textrm{-Hard}$ 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.

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Let (QP)(QP)(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\mathcal{NP}\textrm{-Hard}NP-Hard problems. Standard solvers that use a Branch and Bound algorithm can efficiently solve (QP)(QP)(QP) in the specific case where its objective function is convex. Thus, to solve (QP)(QP)(QP), we choose to reformulate it into an equivalent problem with a convex objective function. Two reformulations are possible: either we reformulate (QP)(QP)(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)(QP)(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)(QP)(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)(QP)(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)(QP)(QP) and that we improve by adding valid inequalities. \par In a second part, we present several quadratic convex reformulations of (QP)(QP)(QP), i.e. we reformulate (QP)(QP)(QP) into an equivalent program, with a quadratic convex objective function. We first introduce a simple approach to convexify (QP)(QP)(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)(QP)(QP). We call this approach \texttt{NC (Naive Convexification)}. Then, we introduce a new convex reformulation scheme that perturbs the objective function of (QP)(QP)(QP) with the linear expression of the products of integer variables, and the equality constraints of (QP)(QP)(QP). Then, we show that we can compute, within this scheme, an optimal reformulation of (QP)(QP)(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)(QP)(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)(QP)(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)}. 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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)(QP)(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 404040 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 problem (QP) of minimizing a quadratic function subject to linear or quadratic constraints. Variables are integer and bounded. This very general problem can model many classical problems in Combinatorial Optimization.A major difference between (QP) and integer linear programs lies in the fact that, in general, its continuous relaxation is an NP-hard optimization problem. To overcome this difficulty, the Convex Quadratic Reformulation approach transforms (QP) into a problem (QP'), equivalent to (QP), but whose continuous relaxation is a convex problem. To compute a global optimal solution to (QP) it becomes possible to solve (QP') by an implicit enumeration algorithm based on continuous convex optimization.We make an overview of recent developments in this approach. We show in particular how positive semidefinite relaxations can be used to build the most interesting equivalent problems (QP'). In the case of binary variables, we give a vision of classical linearization as a special case of Convex Quadratic Reformulation.Finally, we illustrate the generality and the computational efficiency of the Quadratic Convex Reformulation approach for several combinatorial optimization problems.
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.
Let (MQP) be a MIQP that consists in minimizing a quadratic function subject to linear constraints. Our approach to solve (MQP) is first to consider (MQP'), an equivalent MIQP that has a convex objective function, additional variables and constraints, and additionnal quadratic constraints. Then, we propose a new Branch and Bound based on the relaxation of the quadratic constraints to solve (MQP'). We perform experiments on pure-and mixed-integer instances of medium size, and show that their solution times are improved by our Branch and Bound in comparison with two existing approaches.
Nous considérons le problème (QP) de la minimisation d'une fonction quadratique sous des contraintes linéaires ou quadratiques. Les variables sont entières et bornées. Ce problème très général permet de modéliser de nombreux problèmes classiques en Optimisation Combinatoire et constitue une première généralisation de la programmation linéaire en nombres entiers. Une différence majeure entre (QP) et les programmes linéaires en nombres entiers réside dans le fait que, en général, sa relaxation continue fournit un problème lui aussi NP-difficile. Pour contourner cette difficulté, la reformulation quadratique convexe transforme (QP) en un problème (QP') équivalent mais dont la relaxation continue est un problème convexe. Afin de calculer une solution optimale de (QP), on peut alors résoudre (QP') par un algorithme d'énumération implicite basé sur l'optimisation continue convexe. Nous faisons un tour d'horizon de développements récents de cette approche. Nous montrons en particulier comment les relaxations semi-définies positives permettent de construire les problèmes équivalents (QP') les plus intéressants. Dans le cas des variables binaires, nous donnons une vision des linéarisations classiques comme un cas particulier de reformulation quadratique convexe. Enfin, nous illustrons la généralité et l'efficacité expérimentale de la résolution exacte par reformulation quadratique convexe sur différents problèmes d'Optimisation Combinatoire. Références A. Billionnet et S. Elloumi. Using a Mixed Integer Quadratic Programming Solver for the Unconstrained Quadratic 0-1 Problem . Mathematical Programming, 109(1): 55-68, 2007. A. Billionnet, S. Elloumi et M.-C. Plateau. Improving standard solvers convex reformulation for constrained quadratic 0-1 programs: the QCR method. Discrete Applied Math, vol. 157(6), 2009, pp. 1185-1197. A. Billionnet, S. Elloumi et A. Lambert. Extending the QCR method to general mixed-integer programs. Mathematical Programming, vol. 131(1), 2012, pp.381-401
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 combinatorial optimization problems can be formulated as the minimization of a 0–1 quadratic function subject to linear constraints. In this paper, we are interested in the exact solution of this problem through a two-phase general scheme. The first phase consists in reformulating the initial problem either into a compact mixed integer linear program or into a 0–1 quadratic convex program. The second phase simply consists in submitting the reformulated problem to a standard solver. The efficiency of this scheme strongly depends on the quality of the reformulation obtained in phase 1. We show that a good compact linear reformulation can be obtained by solving a continuous linear relaxation of the initial problem. We also show that a good quadratic convex reformulation can be obtained by solving a semidefinite relaxation. In both cases, the obtained reformulation profits from the quality of the underlying relaxation. Hence, the proposed scheme gets around, in a sense, the difficulty to incorporate these costly relaxations in a branch-and-bound algorithm.