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In the context of the maximum stable set problem, rank inequalities impose that the cardinality of any set of vertices contained in a stable set be, at most, as large as the stability number of the subgraph induced by such a set. Rank inequalities are very general, as they subsume many classical inequalities such as clique, hole, antihole, web, and antiweb inequalities. In spite of their generality, the exact separation of rank inequalities has never been addressed without the introduction of topological restrictions on the induced subgraph and the tightness of their closure has never been investigated systematically. In this work, we propose a methodology for optimizing over the closure of all rank inequalities with a right-hand side no larger than a small constant without imposing any restrictions on the topology of the induced subgraph. Our method relies on the exact separation of a relaxation of rank inequalities, which we call relaxed k-rank inequalities, whose closure is as tight. We investigate the corresponding separation problem, a bilevel programming problem asking for a subgraph of maximum weight with a bound on its stability number, whose study could be of independent interest. We first prove that the problem is [Formula: see text]-hard and provide some insights on its polyhedral structure. We then propose two exact methods for its solution: a branch-and-cut algorithm (which relies on a family of faced-defining inequalities which we introduce in this paper) and a purely combinatorial branch-and-bound algorithm. Our computational results show that the closure of rank inequalities with a right-hand side no larger than a small constant can yield a bound that is stronger, in some cases, than Lovász’s Theta function, and substantially stronger than bounds obtained with standard inequalities that are valid for the stable set problem, including odd-cycle inequalities and wheel inequalities. Summary of Contribution: This paper proposes two original methods for solving a challenging cut-separation problem (of bilevel type) for a large class of inequalities valid for one of the key operations research problems, namely, the max stable set problem. An extensive set of experimental results validates the proposed methods. All the source code and data sets are available online on GitHub.

In this paper we address game theory problems arising in the context of network security. In traditional game theory problems, given a defender and an attacker, one searches for mixed strategies which minimize a linear payoff functional. In the problems addressed in this paper an additional quadratic term is added to the minimization problem. Such term represents switching costs, i.e., the costs for the defender of switching from a given strategy to another one at successive rounds of a Nash game. The resulting problems are nonconvex QP ones with linear constraints and turn out to be very challenging. We will show that the most recent approaches for the minimization of nonconvex QP functions over polytopes, including commercial solvers such as and , are unable to solve to optimality even test instances with $$n=50$$ n = 50 variables. For this reason, we propose to extend with them the current benchmark set of test instances for QP problems. We also present a spatial branch-and-bound approach for the solution of these problems, where a predominant role is played by an optimality-based domain reduction, with multiple solutions of LP problems at each node of the branch-and-bound tree. Of course, domain reductions are standard tools in spatial branch-and-bound approaches. However, our contribution lies in the observation that, from the computational point of view, a rather aggressive application of these tools appears to be the best way to tackle the proposed instances. Indeed, according to our experiments, while they make the computational cost per node high, this is largely compensated by the rather slow growth of the number of nodes in the branch-and-bound tree, so that the proposed approach strongly outperforms the existing solvers for QP problems.

In this study, we consider identification of parameters in a non-linear continuum-mechanical model of arteries by fitting the models response to clinical data. The fitting of the model is formulated as a constrained non-linear, non-convex least-squares minimization problem. The model parameters are directly related to the underlying physiology of arteries, and correctly identified they can be of great clinical value. The non-convexity of the minimization problem implies that incorrect parameter values, corresponding to local minima or stationary points may be found, however. Therefore, we investigate the feasibility of using a branch-and-bound algorithm to identify the parameters to global optimality. The algorithm is tested on three clinical data sets, in each case using four increasingly larger regions around a candidate global solution in the parameter space. In all cases, the candidate global solution is found already in the initialization phase when solving the original non-convex minimization problem from multiple starting points, and the remaining time is spent on increasing the lower bound on the optimal value. Although the branch-and-bound algorithm is parallelized, the overall procedure is in general very time-consuming. Funding Agencies|Swedish Research CouncilSwedish Research CouncilEuropean Commission [621-2014-4165]

We consider a nonconvex mixed-integer nonlinear programming (MINLP) model proposed by Goldberg et al. (Comput Optim Appl 58:523–541, 2014. 10.1007/s10589-014-9647-y) for piecewise linear function fitting. We show that this MINLP model is incomplete and can result in a piecewise linear curve that is not the graph of a function, because it misses a set of necessary constraints. We provide two counterexamples to illustrate this effect, and propose three alternative models that correct this behavior. We investigate the theoretical relationship between these models and evaluate their computational performance.

International audience; Finding solutions to least-squares problems with low cardinality has found many applications, including portfolio optimization, subset selection in statistics, and inverse problems in signal processing. Although most works consider local approaches that scale with high-dimensional problems, some others address its global optimization via mixed integer programming (MIP) reformulations. We propose dedicated branch-and-bound methods for the exact resolution of moderate-size, yet difficult, sparse optimization problems, through three possible formulations: cardinality-constrained and cardinality-penalized least-squares, and cardinality minimization under quadratic constraints. A specific tree exploration strategy is built. Continuous relaxation problems involved at each node are reformulated as (Formula presented.) -norm-based optimization problems, for which a dedicated algorithm is designed. The obtained certified solutions are shown to better estimate sparsity patterns than standard methods on simulated variable selection problems involving highly correlated variables. Problem instances selecting up to 24 components among 100 variables, and up to 15 components among 1000 variables, can be solved in less than 1000 s. Unguaranteed solutions obtained by limiting the computing time to 1s are also shown to provide competitive estimates. Our algorithms strongly outperform the CPLEX MIP solver as the dimension increases, especially for quadratically-constrained problems. The source codes are made freely available online.

Of central importance to the $$\alpha $$ BB algorithm is the calculation of the $$\alpha $$ values that guarantee the convexity of the underestimator. Improvement (reduction) of these values can result in tighter underestimators and thus increase the performance of the algorithm. For instance, it was shown by Wechsung et al. (J Glob Optim 58(3):429–438, 2014) that the emergence of the cluster effect can depend on the magnitude of the $$\alpha $$ values. Motivated by this, we present a refinement method that can improve (reduce) the magnitude of $$\alpha $$ values given by the scaled Gerschgorin method and thus create tighter convex underestimators for the $$\alpha $$ BB algorithm. We apply the new method and compare it with the scaled Gerschgorin on randomly generated interval symmetric matrices as well as interval Hessians taken from test functions. As a measure of comparison, we use the maximal separation distance between the original function and the underestimator. Based on the results obtained, we conclude that the proposed refinement method can significantly reduce the maximal separation distance when compared to the scaled Gerschgorin method. This approach therefore has the potential to improve the performance of the $$\alpha $$ BB algorithm.

International audience; This paper introduces the problem of backfeed fiber cables network design. It considers cable separation operations and costs as well as a non-linear cable line cost, and the feedback technique. An integer programming based solution is proposed, and some associated valid inequalities are introduced. The problem is proven to be NP-Hard. The formulation is assessed on real-life instances

In this paper we propose convex and LP bounds for Standard Quadratic Programming (StQP) problems and employ them within a branch-and-bound approach. We first compare different bounding strategies for StQPs in terms both of the quality of the bound and of the computation times. It turns out that the polyhedral bounding strategy is the best one to be used within a branch-and-bound scheme. Indeed, it guarantees a good quality of the bound at the expense of a very limited computation time. The proposed branch-and-bound algorithm performs an implicit enumeration of all the KKT (stationary) points of the problem. We compare different branching strategies exploiting the structure of the problem. Numerical results on randomly generated problems (with varying density of the underlying convexity graph) are reported which show the effectiveness of the proposed approach, in particular in limiting the growth of the number of nodes in the branch-and-bound tree as the density of the underlying graph increases.

A new branch-and-bound algorithm for the maximum weight clique problem is proposed. The proposed algorithm consists of two phases, a precomputation phase and a branch-and-bound phase. In the precomputation phase, the weights of maximum weight cliques in many small subgraphs are calculated and stored in optimal tables. In the branch-and-bound phase, each problem is divided into smaller subproblems, and unnecessary subproblems are pruned using the optimal tables. We performed experiments with the proposed algorithm and five existing algorithms for several types of graphs. The results indicate that only the proposed algorithm can obtain exact solutions for all graphs and that it performs much faster than other algorithms for nearly all graphs.

We introduce a strong extended formulation of the convex recoloring problem on a tree, which has an application in analyzing phylogenetic trees. The extended formulation has only a polynomial number of constraints, but dominates the conventional formulation and the exponentially many valid inequalities introduced by Campelo et al. (Math Progr 156:303–330, 2016). We show that all valid inequalities introduced by Campelo et al. can be derived from the extended formulation. We also show that the natural restriction of the extended formulation provides a complete inequality description of the polytope of subtrees of a tree. The solution time using the extended formulation is much smaller than that with the conventional formulation. Moreover the extended formulation solves all the problem instances attempted in Campelo et al. (2016) and larger sized instances at the root node of the branch-and-bound tree without branching.

This paper studies the robust cyclic timetabling problem. The goal is to modify a given reference timetable to enhance its robustness against small stochastic disturbances. The robustness is measured by the expected total delay of the realised timetable. Kroon et al. (Transp Res Part B 42(6):553–570, 2008) propose a stochastic programming approach and implement it for Netherlands Railways (NS). While the model’s outcome is accepted by practitioners, relevant planning problems are rendered intractable by computation times of up to several days. In this paper we describe a Branch-and-Bound algorithm for solving the stochastic program of Kroon et al. (Transp Res Part B 42(6):553–570, 2008). We propose specific node selection rules, variable selection rules, constructive heuristics and lower bounds. We carry out computational tests on large real-life problem instances. The results confirm that our algorithm is able to considerably improve the robustness of the reference solutions. This is achieved with computation times of a few minutes. However, the weak lower bounds we use leave a considerable optimality gap. Therefore, our algorithm is best described as a heuristic solution method.

International audience; We investigate a variant of the Graph Partitioning Problem with capacity constraints imposed on the clusters, giving rise to quadratic constraints in 0–1 formulations. Several compact linearized models of the problem are proposed and analysed: (a) a model featuring binary variables which results from the application of the standard Fortet linearization technique; (b) a more compact model featuring only binary variables, obtained by linearization after reformulation of the quadratic constraints as bilinear constraints; (c) a strengthened version of the latter model, still featuring variables. Computational experiments comparing the relative strength and efficiency of the various models on a series of test instances involving complete graphs with up to 50 nodes are reported and discussed.

In this paper, we deal with the Time-Dependent Asymmetric Traveling Salesman Problem with Time Windows. First, we prove that under special conditions the problem can be solved as an Asymmetric Traveling Salesman Problem with Time Windows, with suitable-defined time windows and (constant) travel times. Second, we show that, if the special conditions do not hold, the time-independent optimal solution provides both a lower bound and (eventually) an upper bound with a worst-case guarantee for the Time-Dependent Asymmetric Traveling Salesman Problem with Time Windows. Finally, a branch-and-bound algorithm is presented and tested on a set of 4800 instances. The results have been compared with those obtained by the only existing exact procedure capable of solving this problem. The new procedure has been able to find a higher number of optimal solutions in this set of instances.

International audience; We consider the problem of finding a shortest path in a directed graph with a quadratic objective func- tion (the QSPP). We show that the QSPP cannot be approximated unless P = NP . For the case of a convex objective function, an n -approximation algorithm is presented, where n is the number of nodes in the graph, and APX -hardness is shown. Furthermore, we prove that even if only adjacent arcs play a part in the quadratic objective function, the problem still cannot be approximated unless P = NP . In order to solve the problem we first propose a mixed integer programming formulation, and then devise an ef- ficient exact Branch-and-Bound algorithm for the general QSPP, where lower bounds are computed by considering a reformulation scheme that is solvable through a number of minimum cost flow problems. In our computational experiments we solve to optimality different classes of instances with up to 10 0 0 nodes.

This paper studies hub network design problems with profits. They consider a profit-oriented objective that measure the tradeoff between the revenue due to served commodities and the overall network design and transportation costs. An exact algorithmic framework is proposed for two variants of this class of problems, where a sophisticated Lagrangian function that exploits the structure of the problems is used to efficiently obtain bounds at the nodes of an enumeration tree. In addition, reduction tests and partial enumerations are used to considerably reduce the size of the problems and thus help decrease the computational effort. Numerical results on a set of benchmark instances with up to 100 nodes confirm the efficiency of the proposed algorithmic framework. The proposed methodology can be used as a tool to solve more complex variants of this class of problems as well as other discrete location and network design problems involving servicing decisions.

In this paper we discuss optimality-based domain reductions for Global Optimization problems both from the theoretical and from the computational point of view. When applying an optimality-based domain reduction we can easily define a lower limit for the reduction which can be attained, but we can hardly guarantee that such limit is reached. Here, we theoretically prove that, for a nontrivial class of problems, appropriate strategies exist that are always able to reach this lower limit. On the other hand, we will also show that the same strategies lose this property as soon as we slightly enlarge the class of problems. Next, we perform computational experiments with a standard B&B approach applied to Linear Multiplicative Programming problems. We aim at establishing a good trade off between the quality of the domain reduction (the higher the quality, the lower the number of nodes in the B&B tree), and the computational cost of the domain reduction, and, thus, the effort per node of the B&B tree.

The iterative bisection of the longest edge of the unit simplex generates a binary tree, where the specific shape depends on the chosen longest edges to be bisected. In global optimization, the use of various distance norms may be advantageous for bounding purposes. The question dealt with in this paper is how the size of a binary tree generated by the refinement process depends on heuristics for longest edge selection when various distance norms are used. We focus on the minimum size of the tree that can be reached, how selection criteria may reduce the size of the tree compared to selecting the first edge, whether a predefined grid is covered and how unique are the selection criteria. The exact numerical values are provided for the unit simplex in 4 and 5-dimensional space.

Many algorithms for globally solving sum of affine ratios problem (SAR) are based on equivalent problem and branch-and-bound framework. Since the exhaustiveness of branching rule leads to a significant increase in the computational burden for solving the equivalent problem. In this study, a new range reduction method for outcome space of the denominator is presented for globally solving the sum of affine ratios problem (SAR). The proposed range reduction method offers a possibility to delete a large part of the outcome space region of the denominators in which the global optimal solution of the equivalent problem does not exist, and which can be seen as an accelerating device for global optimization of the (SAR). Several numerical examples are presented to demonstrate the advantages of the proposed algorithm using new range reduction method in terms of both computational efficiency and solution quality.

This article presents BiqCrunch , an exact solver for binary quadratic optimization problems. BiqCrunch is a branch-and-bound method that uses an original, efficient semidefinite-optimization-based bounding procedure. It has been successfully tested on a variety of well-known combinatorial optimization problems, such as Max-Cut, Max- k -Cluster, and Max-Independent-Set. The code is publicly available online; a web interface and many conversion tools are also provided.

Backtrack-style exhaustive search algorithms for NP-hard problems tend to have large variance in their runtime. This is because “fortunate” branching decisions can lead to finding a solution quickly, whereas “unfortunate” decisions in another run can lead the algorithm to a region of the search space with no solutions. In the literature, frequent restarting has been suggested as a means to overcome this problem. In this paper, we propose a more sophisticated approach: a best-firstsearch heuristic to quickly move between parts of the search space, always concentrating on the most promising region. We describe how this idea can be efficiently incorporated into a backtrack search algorithm, without sacrificing optimality. Moreover, we demonstrate empirically that, for hard solvable problem instances, the new approach provides significantly higher speed-up than frequent restarting.

The biggest challenge when disclosing private data is to share information contained in databases while protecting people from being individually identified. Microaggregation is a family of methods for statistical disclosure control. The principle of microaggregation is that confidentiality rules permit the publication of individual records if they are partitioned into groups of size larger or equal to a fixed threshold value, where none is more representative than the others in the same group. The application of such rules leads to replacing individual values by those computed from small groups (microaggregates), before data publication. This work proposes a column generation algorithm for numerical microaggregation in which its pricing problem is solved by a specialized branch-and-bound. The algorithm is able to find, for the first time, lower bounds for instances of three real-world datasets commonly used in the literature. Furthermore, new best known solutions are obtained for these instances by means of a simple heuristic method with the columns generated.

In several areas like global optimization using branch-and-bound methods for mixture design, the unit n-simplex is refined by longest edge bisection (LEB). This process provides a binary search tree. For (Formula presented.), simplices appearing during the refinement process can have more than one longest edge (LE). The size of the resulting binary tree depends on the specific sequence of bisected longest edges. The questions are how to calculate the size of one of the smallest binary trees generated by LEB and how to find the corresponding sequence of LEs to bisect, which can be represented by a set of LE indices. Algorithms answering these questions are presented here. We focus on sets of LE indices that are repeated at a level of the binary tree. A set of LEs was presented in Aparicio et al. (Informatica 26(1):17–32, 2015), for (Formula presented.). An additional question is whether this set is the best one under the so-called (Formula presented.)-valid condition.

International audience; We consider a variant of the graph partitioning problem involving knapsack constraints with Gaussian random coefficients. In this new variant, under this assumption of probability distribution, the problem can be traditionally formulated as a binary SOCP for which the continuous relaxation is convex. In this paper, we reformulate the problem as a binary quadratic constrained program for which the continuous relaxation is not necessarily convex. We propose several linearization techniques for latter: the classical linearization proposed by Fortet (Trabajos de Estadistica 11(2):111–118, 1960) and the linearization proposed by Sherali and Smith (Optim Lett 1(1):33–47, 2007). In addition to the basic implementation of the latter, we propose an improvement which includes, in the computation, constraints coming from the SOCP formulation. Numerical results show that an improvement of Sherali–Smith’s linearization outperforms largely the binary SOCP program and the classical linearization when investigated in a branch-and-bound approach.

We present and explore the behaviour of a branch-and-bound algorithm for calculating valid bounds on the k-th largest eigenvalue of a symmetric interval matrix. Branching on the interval elements of the matrix takes place in conjunction with the application of Rohn’s method (an interval extension of Weyl’s theorem) in order to obtain valid outer bounds on the eigenvalues. Inner bounds are obtained with the use of two local search methods. The algorithm has the theoretical property that it provides bounds to any arbitrary precision > 0 (assuming infinite precision arithmetic) within finite time. In contrast with existing methods, bounds for each individual eigenvalue can be obtained even if its range overlaps with the ranges of other eigenvalues. Performance analysis is carried out through nine examples. In the first example, a comparison of the efficiency of the two local search methods is reported using 4,000 randomly generated matrices. The eigenvalue bounding algorithm is then applied to five randomly generated matrices with overlapping eigenvalue ranges. Valid and sharp bounds are indeed identified given a sufficient number of iterations. Furthermore, most of the range reduction takes place in the first few steps of the algorithm so that significant benefits can be derived without full convergence. Finally, in the last three examples, the potential of the algorithm for use in algorithms to identify index-1 saddle points of nonlinear functions is demonstrated.

Given a set of identical capacitated bins, a set of weighted items, and a set of precedences among such items, we are interested in determining the minimum number of bins that can accommodate all items and can be ordered in such a way that all precedences are satisfied. The problem, denoted as the bin packing problem with precedence constraints (BPP-P), has a very intriguing combinatorial structure and models many assembly and scheduling issues. According to our knowledge, the BPP-P has received little attention in the literature, and in this paper we address it for the first time with exact solution methods. In particular, we develop reduction criteria, a large set of lower bounds, a variable neighborhood search upper bounding technique, and a branch-and-bound algorithm. We show the effectiveness of the proposed algorithms by means of extensive computational tests on benchmark instances and comparison with standard integer linear programming techniques.