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Theses Year : 2009

## Résolution de programmes quadratiques en nombres entiers

Amélie Lambert

#### 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.

#### Domains

Operations Research [cs.RO]

### Dates and versions

tel-02459253 , version 1 (29-01-2020)

### Identifiers

• HAL Id : tel-02459253 , version 1

### Cite

Amélie Lambert. Résolution de programmes quadratiques en nombres entiers. Recherche opérationnelle [cs.RO]. Conservatoire National des Arts et Métiers (CNAM), 2009. Français. ⟨NNT : ⟩. ⟨tel-02459253⟩

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