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

(Global) Optimization: Historical notes and recent developments

Recent developments in (Global) Optimization are surveyed in this paper. We collected and commented quite a large number of recent references which, in our opinion, well represent the vivacity, deepness, and width of scope of current computational approaches and theoretical results about nonconvex optimization problems. Before the presentation of the recent developments, which are subdivided into two parts related to heuristic and exact approaches, respectively, we briefly sketch the origin of the discipline and observe what, from the initial attempts, survived, what was not considered at all as well as a few approaches which have been recently rediscovered, mostly in connection with machine learning

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]

Combinatorial optimization under ellipsoidal uncertainty

We study combinatorial problems with ellipsoidal uncertainty in the objective function concerning their theoretical and practical solvability. Ellipsoidal uncertainty is a natural model when the coefficients are normally distributed random variables. Robust versions of typical combinatorial problems can be very hard to solve compared to their linear versions. Complexity and approaches differ fundamentally depending on whether uncorrelated or correlated uncertainty occurs. We distinguish between these two cases and consider first the unconstrained binary optimization under uncorrelated ellipsoidal uncertainty. For this we develop an algorithm which computes an optimal solution by merely sorting the variables and, correspondingly, has a running time of O(n log n). The algorithm is based on the diminishing returns-property, which is characteristic for submodular functions. We introduce a new and a more general p-norm-uncertainty and show that with only slight modifications the sorting algorithm can be easily applied. We also extend the algorithm to general integer variables, which in this case only leads to a pseudo-polynomial time. The next step to the general case is investigation of problems with arbitrary combinatorial sets X ⊆ {0, 1}n under uncorrelated ellipsoidal uncertainty. For this case we embed the O(n log n)-algorithm for the unconstrained binary problems into a Lagrangean decomposition approach. The approach separates the objective function from the combinatorial structure applying Lagrangean relaxation to some artificial connecting constraints. This creates two subproblems, one of which is the linear version of the combinatorial problem and the other one is just the unconstrained binary uncorrelated problem, which can be solved using the O(n log n)-algorithm. The solutions of the subproblems are used to obtain primal and dual bounds which are used in a branch and bound-approach. The approach shows an excellent performance in practice. In the correlated case already the unconstrained binary problem turns out to be strongly NP-hard. Here we also define a branch and bound-approach, now with lower bounds determined by underestimation of the given ellipsoid with certainly defined axis-parallel ellipsoids. We use this idea to extend the decomposition approach to general combinatorial problems under correlated uncertainty. In contrast to the uncorrelated case the uncertain subproblem of the decomposition is here strongly NP-hard in itself. We solve it approximately using the developed underestimators which are determined in a preprocessing step. The approach offers room for improvement concerning in the primal extent a faster computation of the underestimators, which is done by solving semidefinite programs