In this paper we introduce new semidefinite programming relaxations to box-constrained polynomial optimization programs (P). For this, we first reformulate (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.

Optimal Power Flow (OPF) can be modeled as a nonconvex Quadratically Constrained Quadratic Program (QCQP). Our purpose is to solve OPF to global optimality. To this end, we specialize the Mixed-Integer Quadratic Convex Reformulation method (MIQCR) to (OPF). This is a method in two steps. First, a Semi-Definite Programming (SDP) relaxation of (OPF) is solved. Then the optimal dual variables of this relaxation are used to reformulate OPF into an equivalent new quadratic program, where all the non-convexity is moved to one additional constraint. In the second step, this reformulation is solved within a branch-and-bound algorithm, where at each node a quadratic and convex relaxation of the reformulated problem, obtained by relaxing the non-convex added constraint, is solved. The key point of our approach is that the lower bound at the root node of the branch-and-bound tree is equal to the SDP relaxation value. We test this method on several OPF cases, from two-bus networks to more-than-a-thousand-buses networks from the MAT-POWER repository. Our first results are very encouraging.

We present algorithm MIQCR-CB that is an advancement of MIQCR. 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); in the second phase, the equivalent formulation is solved by a standard solver. Because the reformulation relies on the solution of a large-scale semidefinite program, it is not tractable by existing semidefinite solvers even 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. First, we apply MIQCR-CB to the k-cluster problem that can be formulated by a binary quadratic program. As an illustration of the efficiency of our new algorithm, for instances of size 80 and of density 25%, MIQCR-CB is on average 78 times faster for phase 1 and 24 times faster for phase 2 than the original MIQCR. We also compare MIQCR-CB with QCR and with BiqCrunch, two methods devoted to binary quadratic programming. We show that MIQCR-CB is able to solve most of the 225 considered instances within three hours of CPU time. We also present experiments on two classes of general integer instances where we compare MIQCR-CB with MIQCR, Couenne, and Cplex12.6. We demonstrate the significant improvement over the original MIQCR approach. Finally, we show that MIQCR-CB is able to solve almost all of the considered instances, whereas Couenne and Cplex12.6 are not able to solve half of them.