The study deals with minimization of a multiple-extremum quadratic functional defined in the state space of binary variables. In order to speed the computation of the local field (which is similar to the gradient in the continuous space), we offer binarization of the matrix on which the functional is built by substituting its elements with 0, +1, or −1 according to their signs. We show that the binarization can be performed so that the computed and real directions of the local field will very likely be the same. The algorithm is particularly useful for treating high-dimensionality configuration spaces because the binarization of the matrix considerably simplifies computations and decreases the amount of memory needed.
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 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.
Practical discrete optimization problems often contain multidimensional arrays of variables interrelated by linear constraints such as equalities and inequalities. Values of each variable depend on its specific meaning and can be binary, integer, or discrete. These conditions make it technically difficult to reduce the original problem statement to QUBO form. We identify and examine three necessary transformations of the original problem statement to reduce it to QUBO form, namely transition from a multidimensional to a one-dimensional array, transition to binary variables in mixed problems, and incorporating linear constraints into the objective function in the form of quadratic penalties. We present and prove computationally convenient formulas to simplify these transformations. In particular, the formulas for the transition from a multidimensional to a one-dimensional array of variables are based on the application of the Kronecker product of matrices. The transformations considered are illustrated by numerous examples and used, as an application, to reduce a number of well-known problems in graph theory and combinatorial optimization to QUBO form.
In this paper the problem or maximizing a quadratic function defined in {-l, l}^n is considered. We propose a technique to obtain an upper bound and a lower bound to the maximum of a quadratic function on the set {-l, l}^n and a feasible point where the lower bound is attained. The problem of the approximability of the quadratic maximization problem with integer constraints by the method proposed here is studied and solved negatively. oreover a special class \nof matrices such that the feasible point obtained with our method is the solution \nor the maximization problem considered is given. Numerical implementation of the \nmethod proposed and related numerical experience are shown.