Abstract We propose a two‐stage recoverable robustness approach that minimizes the recovery cost. In many applications, once the uncertainty is revealed, it can be more important to recover a solution which is as similar as possible to the nominal solution than to minimize the nominal objective value of . This for example occurs when the nominal solution is implemented on a regular basis or when the uncertainty is revealed late. We define the proactive problem which minimizes the weighted recovery costs over a discrete set of scenarios while ensuring optimality of the nominal objective value of . We model the recovery cost of a scenario by a distance between the first‐stage nominal solution and the second‐stage solution recovered for this scenario. We show for two different solution distances and that the proactive problem is ‐hard for both the integer min‐cost flow problem with uncertain arc demands and for the integer max‐flow problem with uncertain arc capacities. For these two problems, we prove that once uncertainty is revealed, even identifying a reactive solution with a minimal distance to a given solution is ‐hard for , and is polynomial for . We highlight the benefits of the proactive approach in a case study on a railroad planning problem. First, we compare it to the anchored and the ‐distance approaches. Then, we show the efficiency of the proactive solution over reactive solutions. Finally, we illustrate the recovery cost reduction when relaxing the optimality constraint on the nominal objective of the proactive solution . We also consider the min–max version of the proactive problem where we minimize the maximal recovery cost over all scenarios. We show that the same complexity results hold for this version. We also exhibit a class of problems for which the set of extreme points of the convex hull of a discrete uncertainty set always contain a worst‐case scenario. We show that this result does not hold for three distinct classes deduced from the first one.

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.

Wireless sensor networks have been widely deployed in the last decades to provide various services, like environmental monitoring or object tracking. Such a network is composed of a set of sensor nodes which are used to sense and transmit collected information to a base station. To achieve this goal, two properties have to be guaranteed: (i) the sensor nodes must be placed such that all the environment of interest is covered, and (ii) every sensor node can transmit its data to the base station (through other sensor nodes). In this paper, we consider the Minimum Connected Coverage (MCC) problem. We propose two mathematical programming formulations for the MCC problem on square grid graphs. We compare them to a recent model proposed by Rebai et al [1]. Our mathematical programming formulations yield a better LP-bound at the root of the branch-and-cut process than the model of Rebai et al. Moreover, the presented formulations outperform the proportion of solved instances in their work as well as the CPU computation time and the number of nodes explored in the tree search.

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.

Abstract Optimality of a linear inequality in finitely many graph invariants is defined through a geometric approach. For a fixed number of graph vertices, consider all the tuples of values taken by the invariants on a selected class of graphs. Then form the polytope which is the convex hull of all these tuples. By definition, the optimal linear inequalities correspond to the facets of this polytope. They are finite in number, are logically independent, and generate precisely all the linear inequalities valid on the class of graphs. The computer system GraPHedron, developed by some of the authors, is able to produce experimental data about such inequalities for a “small” number of vertices. It greatly helps in conjecturing optimal linear inequalities, which are then hopefully proved for any number of vertices. Two examples are investigated here for the class of connected graphs. First, all the optimal linear inequalities for the stability number and the number of edges are obtained. To this aim, a problem of Ore (1962) related to the Turán Theorem (1941) is solved. Second, several optimal inequalities are established for three invariants: the maximum degree, the irregularity, and the diameter. © 2008 Wiley Periodicals, Inc. NETWORKS, 2008

The p-center problem consists of choosing p facilities among a set of M possible locations and assigning N clients to them in order to minimize the maximum distance between a client and the facility to which it is allocated. We present a new integer linear programming formulation for this min-max problem with a polynomial number of variables and constraints, and show that its LP relaxation provides a lower bound tighter than the classical one. Moreover, we show that an even better lower bound LB*, obtained by keeping the integrality restrictions on a subset of the variables, can be computed in polynomial time by solving at most O(log2(NM)) linear programs, each having N rows and M columns. We also show that, when the distances satisfy triangle inequalities, LB* is at least one third of the optimal value. Finally, we use LB* in an exact solution method and report extensive computational results on test problems from the literature. For instances where the triangle inequalities are satisfied, our method outperforms the running time of other recent exact methods by an order of magnitude. Moreover, it is the first one to solve large instances of size up to N = M = 1,817.