Using a conic bundle method to accelerate both phases of a quadratic convex reformulation

We present algorithm MIQCR-CB that is an advancement of method MIQCR~(Billionnet, Elloumi and Lambert, 2012). MIQCR is a method for solving mixed-integer quadratic programs and works in two phases: the first phase determines an equivalent quadratic formulation with a convex objective function by solving a semidefinite problem $(SDP)$, and, in the second phase, the equivalent formulation is solved by a standard solver. As the reformulation relies on the solution of a large-scale semidefinite program, it is not tractable by existing semidefinite solvers, already for medium sized problems. To surmount this difficulty, we present in MIQCR-CB a subgradient algorithm within a Lagrangian duality framework for solving $(SDP)$ that substantially speeds up the first phase. Moreover, this algorithm leads to a reformulated problem of smaller size than the one obtained by the original MIQCR method which results in a shorter time for solving the second phase. We present extensive computational results to show the efficiency of our algorithm

Global solution of non-convex quadratically constrained quadratic programs

International audienceThe class of mixed-integer quadratically constrained quadratic programs (QCQP) consists of minimizing a quadratic function under quadratic constraints where the variables could be integer or continuous. On a previous paper we introduced a method called MIQCR for solving QC-QPs with the following restriction : all quadratic sub-functions of purely continuous variables are already convex. In this paper, we propose an extension of MIQCR which applies to any QCQP. Let (P) be a QCQP. Our approach to solve (P) is first to build an equivalent mixed-integer quadratic problem (P *). This equivalent problem (P *) has a quadratic convex objective function, linear constraints, and additional variables y that are meant to satisfy the additional quadratic constraints y = xx T , where x are the initial variables of problem (P). We then propose to solve (P *) by a branch-and-bound algorithm based on the relaxation of the additional quadratic constraints and of the integrality constraints. This type of branching is known as spatial branch-and-bound. Computational experiences are carried out on a total of 325 instances. The results show that the solution time of most of the considered instances is improved by our method in comparison with the recent implementation of QuadProgBB, and with the solvers Cplex, Couenne, Scip, BARON and GloMIQO

Solving a general mixed-integer quadratic problem through convex reformulation : a computational study

International audienceLet (QP) be a mixed integer quadratic program that consists of minimizing a qua-dratic function subject to linear constraints. In this paper, we present a convex reformulation of (QP), i.e. we reformulate (QP) into an equivalent program, with a convex objective function. Such a reformulation can be solved by a standard solver that uses a branch and bound algorithm. This reformulation, that we call MIQCR (Mixed Integer Quadratic Convex Reformulation), is the best one within a convex reformulation scheme, from the continuous relaxation point of view. It is based on the solution of an SDP relaxation of (QP). Computational experiences were carried out with instances of (QP) with one equality constraint. The results show that most of the considered instances, with up to 60 variables, can be solved within 1 hour of CPU time by a standard solver

Semidefinite programming relaxations through quadratic reformulation for box-constrained polynomial optimization problems

International audienceIn this paper we introduce new semidefinite programming relaxations to box-constrained polynomial optimization programs (P). For this, we first reformu-late (P) into a quadratic program. More precisely, we recursively reduce the degree of (P) to two by substituting the product of two variables by a new one. We obtain a quadratically constrained quadratic program. We build a first immediate SDP relaxation in the dimension of the total number of variables. We then strengthen the SDP relaxation by use of valid constraints that follow from the quadratization. We finally show the tightness of our relaxations through several experiments on box polynomial instances

Solving unconstrained 0-1 polynomial programs through quadratic convex reformulation

We propose a solution approach for the problem (P) of minimizing an unconstrained binary polynomial optimization problem. We call this method PQCR (Polynomial Quadratic Convex Reformulation). The resolution is based on a 3-phase method. The first phase consists in reformulating (P) into a quadratic program (QP). For this, we recursively reduce the degree of (P) to two, by use of the standard substitution of the product of two variables by a new one. We then obtain a linearly constrained binary program. In the second phase, we rewrite the quadratic objective function into an equivalent and parametrized quadratic function using the equality x 2 i = x i and new valid quadratic equalities. Then, we focus on finding the best parameters to get a quadratic convex program which continuous relaxation's optimal value is maximized. For this, we build a semidefinite relaxation (SDP) of (QP). Then, we prove that the standard linearization inequalities, used for the quadratization step, are redundant in (SDP) in presence of the new quadratic equalities. Next, we deduce our optimal parameters from the dual optimal solution of (SDP). The third phase consists in solving (QP *), the optimal reformulated problem, with a standard solver. In particular, at each node of the branch-and-bound, the solver computes the optimal value of a continuous quadratic convex program. We present computational results on instances of the image restoration problem and of the low autocorrelation binary sequence problem. We compare PQCR with other convexification methods, and with the general solver Baron 17.4.1 [39]. We observe that most of the considered instances can be solved with our approach combined with the use of Cplex [24]

Quadratization and convexification in polynomial binary optimization

In this paper, we discuss several reformulations and solution approaches for the problem of minimizing a polynomial in binary variables UBPP. We review and integrate different literature streams to describe a methodology consisting of three distinct phases, together with several possible variants for each phase. The first phase determines a recursive decomposition of each monomial of interest into pairs of submonomials, down to the initial variables. The decomposition gives rise to a so-called quadratization scheme. The second phase builds a quadratic reformulation of UBPP from a given quadratization scheme, by associating a new auxiliary variable with each submonomial that appears in the scheme. A quadratic reformulation of UBPP is obtained by enforcing relations between the auxiliary variables and the monomials that they represent, either through linear constraints or through penalty terms in the objective function. The resulting quadratic problem QBP is non-convex in general and is still difficult to solve. At this stage we introduce the third phase of the resolution process, which consists in convexifying QBP. We consider different types of convexification methods, including complete linearization or quadratic convex reformulations. Theoretical properties of the different phases are recalled from the literature or are further clarified. Finally, we present some experimental results to illustrate the discussion

Generation of the Brucella melitensis ORFeome version 1.1.

The bacteria of the Brucella genus are responsible for a worldwide zoonosis called brucellosis. They belong to the alpha-proteobacteria group, as many other bacteria that live in close association with a eukaryotic host. Importantly, the Brucellae are mainly intracellular pathogens, and the molecular mechanisms of their virulence are still poorly understood. Using the complete genome sequence of Brucella melitensis, we generated a database of protein-coding open reading frames (ORFs) and constructed an ORFeome library of 3091 Gateway Entry clones, each containing a defined ORF. This first version of the Brucella ORFeome (v1.1) provides the coding sequences in a user-friendly format amenable to high-throughput functional genomic and proteomic experiments, as the ORFs are conveniently transferable from the Entry clones to various Expression vectors by recombinational cloning. The cloning of the Brucella ORFeome v1.1 should help to provide a better understanding of the molecular mechanisms of virulence, including the identification of bacterial protein-protein interactions, but also interactions between bacterial effectors and their host's targets