We consider the NP-hard problem of minimizing a separable concave quadratic function over the integral points in a polyhedron, and we denote by
Δ the largest absolute value of the subdeterminants of the constraint matrix. In this paper we give an algorithm that finds an
ϵ-approximate solution for this problem by solving a number of integer linear programs whose constraint matrices have subdeterminants bounded by
Δ in absolute value. The number of these integer linear programs is polynomial in the dimension
n, in
Δ, and in
1/ϵ, provided that the number
k of variables that appear nonlinearly in the objective is fixed. As a corollary, we obtain the first polynomial-time approximation algorithm for separable concave integer quadratic programming with
Δ≤2 and
k fixed. In the totally unimodular case
Δ=1, we give an improved algorithm that only needs to solve a number of linear programs that is polynomial in
1/ϵ and is independent of
n, provided that
k is fixed.
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 present an algorithm that finds an epsilon-approximate solution to a mixed integer quadratic programming (MIQP) problem, and that runs on a Turing machine in time polynomial in the size of the instance and in 1/epsilon, provided that the number of integer variables and the number of negative eigenvalues of the Hessian of the objective function are fixed. Unless P=NP, both restrictions are necessary, so this completes the characterization of the complexity of approximating MIQP in terms of the number of integer variables and the inertia of the Hessian; the result is new already in the purely continuous case. The main ingredient is a polynomial-time simultaneous diagonalization algorithm: it computes a rational change of basis that maps a given ellipsoid, presented in factored form, exactly to a ball, while making the objective function separable up to an arbitrarily small perturbation and preserving the inertia of its Hessian. The classical construction, in which the objective function is made exactly separable, requires a change of basis that is in general irrational, and cannot be carried out on a Turing machine; ours rests instead on rational Jacobi rotations, which we believe to be of independent interest.
Solving convex quadratic integer minimization problems by a branch-and-bound algorithm requires tight lower bounds on the optimal objective value. To obtain such dual bounds, we follow the approach of [C. Buchheim, A. Caprara, and A. Lodi, Math. Program., 135 (2012), pp. 369--395] and underestimate the objective function by another convex quadratic function that can be minimized over the integers by simply rounding its continuous minimizer. In geometric terms, we approximate the sublevel sets of the original objective function by auxiliary ellipsoids having the strong rounding property, introduced by [R. Hübner and A. Schöbel, European J. Oper. Res., 237 (2014), pp. 404--410]. We first consider axisparallel ellipsoids (corresponding to separable convex quadratic functions) and show how to efficiently compute an axisparallel ellipsoid yielding the tightest lower bound, both in the situation where the location of the continuous minimizer is given and in the situation where it varies, as it happens in a branch-and-bound approach. In the latter case, we compute both worst-case and average-case optimal ellipsoids. Moreover, we consider the class of quasi-round ellipsoids (from Hübner and Schöbel). In this case, we do not know whether optimal auxiliary ellipsoids can be computed efficiently. However, we present a heuristic for computing an appropriate quasi-round ellipsoid that still yields valid dual bounds; it turns out that using this approach within a branch-and-bound scheme outperforms the approaches based on axisparallel ellipsoid in terms of running times. We finally combine both approaches by introducing the concept of quasi-axisparallel ellipsoids and present a corresponding extension of the heuristic bound computation.
Binary programs with a quadratic objective function are NP-hard in general, even if the linear optimization problem over the same feasible set is tractable. In this paper, we address such problems by computing quadratic global underestimators of the objective function that are separable but not necessarily convex. Exploiting the binary constraint on the variables, a minimizer of the separable underestimator over the feasible set can be computed by solving an appropriate linear minimization problem over the same feasible set. Embedding the resulting lower bounds into a branch-and-bound framework, we obtain an exact algorithm for the original quadratic binary program. The main practical challenge is the fast computation of an appropriate underestimator, which in our approach reduces to solving a series of semidefinite programs. We exploit the special structure of the resulting problems to obtain a tailored coordinate-descent method for their solution. Our extensive experimental results on various quadratic combinatorial optimization problems show that our approach outperforms both CPLEX and the related QCR method as well as the SDP-based software BiqCrunch on instances of the quadratic shortest path problem and the quadratic assignment problem.