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.
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.
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
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.
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.
The problem of minimizing a pseudo-Boolean function with no additional constraints arises in a variety of applications.A quadratization is a quadratic reformulation of the nonlinear problem obtained by introducing a set of auxiliary binary variables which can be optimized using quadratic optimization techniques.Using the well-known result that a pseudo-Boolean function can be uniquely expressed as a multilinear polynomial, any pseudo-Boolean function can be quadratized by providing a quadratization for negative monomials and one for positive monomials.A desirable property for a quadratization is to introduce small number of auxiliary variables.For the case of negative monomials there exists a quadratization using a single auxiliary variable which is, in addition, submodular.However, much less is known for positive monomials.The best lower bound on the number of variables required by a quadratization in the literature is n-1 2 .We present here new quadratizations of the positive monomial that significantly improve this lower bound decreasing it to a logarithmic bound.This lower bound is derived from a more general result, stating that a quadratization with a logarithmic number of variables can be defined for exact kout-of-n functions, which is a more general class of symmetric functions.Moreover, for exact k-out-of-n functions we also prove that a logarithmic number of variables is necessary when requiring certain symmetry conditions to the quadratization.Finally, we provide quadratizations for general symmetric functions using O( √ n) auxiliary variables.This upper bound nicely matches a lower bound of Ω( √ n) variables that was recently introduced.