Assessment of a non-adaptive deterministic global optimization algorithm for problems with low-dimensional non-convex subspaces

The optimum and at least one optimizing point for convex nonlinear programs can be approximated well by the solution to a linear program (a fact long used in branch and bound algorithms). In more general problems, we can identify subspaces of ‘non-convex variables’ such that, if these variables have sufficiently small ranges, the optimum and at least one optimizing point can be approximated well by the solution of a single linear program. If these subspaces are low-dimensional, this suggests subdividing the variables in the subspace a priori, then producing and solving a fixed, known number of linear programs to obtain an approximation to the solution. The total amount of computation is much more predictable than that required to complete a branch and bound algorithm, and the scheme is ‘embarrassingly parallel’, with little need for either communication or load balancing. We compare such a non-adaptive scheme experimentally to our GlobSol branch and bound implementation, on those problems from the COCONUT project Lib1 test set with non-convex subspaces of dimension four or less, and we discuss potential alterations to both the non-adaptive scheme and our branch and bound process that might change the scope of applicability.     

An improved branch‐and‐bound clustering approach for data partitioning

In this paper, we are concerned with clustering algorithms for vertical partitioning. In particular, we examine the use of a branch‐and‐bound scheme. An existing algorithm using such a scheme may produce infeasible solutions to some problems. We adopt the same branch‐and‐bound scheme and develop a new branching strategy to avoid infeasibility. Illustrative examples are used to demonstrate the effectiveness of our new approach. In addition, we also show how to formulate the horizontal partitioning problem such that the same algorithm can be applied.     

A 2-phase algorithm for solving the single allocation p-hub center problem

The single allocation p-hub center problem is an NP-hard location–allocation problem which consists of locating hub facilities in a network and allocating non-hub nodes to hub nodes such that the maximum distance/cost between origin–destination pairs is minimized. In this paper we present an exact 2-phase algorithm where in the first phase we compute a set of potential optimal hub combinations using a shortest path based branch and bound. This is followed by an allocation phase using a reduced sized formulation which returns the optimal solution. In order to get a good upper bound for the branch and bound we developed a heuristic for the single allocation p-hub center problem based on an ant colony optimization approach. Numerical results on benchmark instances show that the new solution approach is superior over traditional MIP-solver like CPLEX. As a result we are able to provide new optimal solutions for larger problems than those reported previously in literature. We are able to solve problems consisting of up to 400 nodes in reasonable time. To the best of our knowledge these are the largest problems solved in the literature to date.     

Low nonconvexity-rank bilinear matrix inequalities: algorithms and applications in robust controller and structure designs

A branch and bound (BB) algorithm for solving a general class of bilinear matrix inequality (BMI) problems is proposed. First, linear matrix inequality (LMI) constraints are incorporated into BMI constraints in a special way to take advantage of useful information on nonconvex terms. Then, the nonconvexity of the BMI is centralized in coupling constraints so that when the latter are omitted, we get a relaxed LMI problem for computing lower bounds. As in our previous developments, the branching is performed in a reduced dimensional space of complicating variables. This makes the approach practical even with a large dimension of overall variables. Applications of the algorithm to several test problems of robust control are discussed.     

Revisiting Branch and Bound Search Strategies for Machine Scheduling Problems

In the design of exact methods for NP-hard machine scheduling problems, branch and bound algorithms have always been widely considered. In this work we revisit the classic search strategies for branch and bound schemes. We consider a systematic application of the well known dynamic programming dominance property for machine scheduling problems. Several conditions concerning the application of the proposed property with respect to best first, depth first, breadth first search strategies and problem characteristics are presented. Computational testing on single machine and flow shop problems validate in practice the efficiency of the considered approach and suggest that the traditional choice of depth first search with respect to best first and breadth first is strongly questionable.     

Computational aspects of a branch and bound algorithm for quadratic zero-one programming

In this paper we describe computational experience in solving unconstrained quadratic zero-one problems using a branch and bound algorithm. The algorithm incorporates dynamic preprocessing techniques for forcing variables and heuristics to obtain good starting points. Computational results and comparisons with previous studies on several hundred test problems with dimensions up to 200 demonstrate the efficiency of our algorithm.     

Convergent tree-reweighted message passing for energy minimization

Algorithms for discrete energy minimization are of fundamental importance in computer vision. In this paper, we focus on the recent technique proposed by Wainwright et al. [33]—tree-reweighted max-product message passing (TRW). It was inspired by the problem of maximizing a lower bound on the energy. However, the algorithm is not guaranteed to increase this bound—it may actually go down. In addition, TRW does not always converge. We develop a modification of this algorithm which we call sequential tree-reweighted message passing. Its main property is that the bound is guaranteed not to decrease. We also give a weak tree agreement condition which characterizes local maxima of the bound with respect to TRW algorithms. We prove that our algorithm has a limit point that achieves weak tree agreement. Finally, we show that, our algorithm requires half as much memory as traditional message passing approaches. Experimental results demonstrate that on certain synthetic and real problems, our algorithm outperforms both the ordinary belief propagation and tree-reweighted algorithm in [33]. In addition, on stereo problems with Potts interactions, we obtain a lower energy than graph cuts.     

Generation of polynomial-time algorithms for some optimization problems on tree-decomposable graphs

Recent work in the design of efficient algorithms for optimization problems on treedecomposable graphs concentrates on developing general approaches which lead to families of related algorithms, rather than on developing isolatedad hoc algorithms. This paper develops new general approaches to obtain two new families of related polynomial-time algorithms for (1) packing, partitioning, and covering problems and (2) multiset and multiproperty problems. These problems have not been handled by previous general approaches.     

A Branch and Bound Algorithm for Bi-level Discrete Network Design Problem

Discrete network design problem (DNDP) is generally formulated as a bi-level programming. Because of non-convexity of bi-level formulation of DNDP which stems from the equilibrium conditions, finding global optimal solutions are very demanding. In this paper, a new branch and bound algorithm being able to find exact solution of the problem is presented. A lower bound for the upper-level objective and its computation method are developed. Numerical experiments show that our algorithm is superior to previous algorithms in terms of both computation time and solution quality. The conducted experiments indicate that in most cases the first incumbent solution which is obtained within a few seconds is superior to the final solution of some of previous algorithms.     

Arcs-states models for the vehicle routing problem with time windows and related problems

This paper presents several Arcs-States models that can be applied to numerous vehicle routing problems, one of which is the well-known vehicle routing problem with capacities and time windows. In these models, the variables correspond to the states (i.e. the resource quantities) of the vehicles when they travel through an arc. The LP relaxation of the problem provides a lower bound that can be embedded in a branch and bound algorithm that solves the problem exactly. However, for the pseudo-polynomial number of variables and constraints of our models, column and row generations have to be used. Generally, in a branch and bound algorithm, the lower bound needs to be very efficient to minimize the size of the branch and bound trees as far as possible. One of the models we present, the time-only, relies on a relaxation of the Arcs-States model where a resource is removed from the states in the variables. Although the quality of the bounds decreases, the global resolution time is greatly improved, as illustrated on instances of Solomon's benchmark.     

Reoptimization in Lagrangian methods for the 0-1 quadratic knapsack problem

The 0-1 quadratic knapsack problem consists of maximizing a quadratic objective function subject to a linear capacity constraint. To exactly solve large instances of this problem with a tree search algorithm (e.g., a branch and bound method), the knowledge of good lower and upper bounds is crucial for pruning the tree but also for fixing as many variables as possible in a preprocessing phase. The upper bounds used in the best known exact approaches are based on Lagrangian relaxation and decomposition. It appears that the computation of these Lagrangian dual bounds involves the resolution of numerous 0-1 linear knapsack subproblems. Thus, taking this huge number of resolutions into account, we propose to embed reoptimization techniques for improving the efficiency of the preprocessing phase of the 0-1 quadratic knapsack resolution. Namely, reoptimization is introduced to accelerate each independent sequence of 0-1 linear knapsack problems induced by the Lagrangian relaxation as well as the Lagrangian decomposition. Numerous numerical experiments validate the relevance of our approach.     

Optimal reachability for multi-priced timed automata

In this paper, we prove the decidability of the minimal and maximal reachability problems for multi-priced timed automata, an extension of timed automata with multiple cost variables evolving according to given rates for each location. More precisely, we consider the problems of synthesizing the minimal and maximal costs of reaching a given target location. These problems generalize conditional optimal reachability, i.e., the problem of minimizing one primary cost under individual upper bound constraints on the remaining, secondary, costs, and the problem of maximizing the primary cost under individual lower bound constraints on the secondary costs. Furthermore, under the liveness constraint that all traces eventually reach the goal location, we can synthesize all costs combinations that can reach the goal.

The decidability of the minimal reachability problem is proven by constructing a zone-based algorithm that always terminates while synthesizing the optimal cost tuples. For the corresponding maximization problem, we construct two zone-based algorithms, one with and one without the above liveness constraint. All algorithms are presented in the setting of two cost variables and then lifted to an arbitrary number of cost variables.     

Upper and Lower Bounds for Selection on the Mesh

A distance-optimal algorithm for selection on the mesh has proved to be elusive, although distance-optimal algorithms for the related problems of routing and sorting have recently been discovered. In this paper we explain, using the notion of adaptiveness, why techniques used in the currently best selection algorithms cannot lead to a distance-optimal algorithm. For worst-case inputs we apply new techniques to improve the previous best upper bound of 1.22n of Kaklamanis et al. [7] to 1.15n . This improvement is obtained in part by increasing the adaptiveness of previous algorithms. Received May 25, 1995; revised June 1, 1996.     

Reduction strategies and exact algorithms for the disjunctively constrained knapsack problem

In this paper, we optimally solve the disjunctively constrained knapsack problem (DCKP), which is a variant of the standard knapsack problem with special disjunctive constraints. First, we develop a generic exact approach which can be considered as a three-phase procedure. The first phase tries to estimate a starting lower bound. The second phase applies a reduction procedure, combined with the lower bound, in order to fix some decision variables to the optimum. The third phase uses an exact branch and bound algorithm that identifies the optimal values of the free decision variables. Second, we design a specialized exact algorithm based upon a dichotomous search combined with a reduction procedure. Third and last, we propose a modified dichotomous search version which is based upon constructing an equivalent model to the DCKP, adding some dominating constraints, and injecting the so-called covering cut. We evaluate the performance of all versions of the algorithm and compare the obtained results to those of other exact algorithms of the literature. Encouraging results have been obtained.     

Local search with constraint propagation and conflict-based heuristics

Search algorithms for solving csp (Constraint Satisfaction Problems) usually fall into one of two main families: local search algorithms and systematic algorithms. Both families have their advantages. Designing hybrid approaches seems promising since those advantages may be combined into a single approach. In this paper, we present a new hybrid technique. It performs a local search over partial assignments instead of complete assignments, and uses filtering techniques and conflict-based techniques to efficiently guide the search. This new technique benefits from both classical approaches: a priori pruning of the search space from filtering-based search and possible repair of early mistakes from local search. We focus on a specific version of this technique: tabu decision-repair. Experiments done on open-shop scheduling problems show that our approach competes well with the best highly specialized algorithms.     

Lambda-opt neural approaches to quadratic assignment problems

In this article, we propose new analog neural approaches to combinatorial optimization problems, in particular, quadratic assignment problems (QAPs). Our proposed methods are based on an analog version of the lambda-opt heuristics, which simultaneously changes assignments for lambda elements in a permutation. Since we can take a relatively large lambda value, our new methods can achieve a middle-range search over possible solutions, and this helps the system neglect shallow local minima and escape from local minima. In experiments, we have applied our methods to relatively large-scale (N = 80 - 150) QAPs. Results have shown that our new methods are comparable to the present champion algorithms; for two benchmark problems, they are obtain better solutions than the previous champion algorithms.     

非凸二次规划全局极小问题的新型分枝定界算法

针对求解多面集上二次函数的全局近似最优解问题,利用逐步缩小对偶间隙的处理办法,提出了一个新型分枝定界算法。新算法的主要改进之处是利用了Lagrange 对偶性获取下界。最后,用构造和随机产生的问题实例,对提出的新算法和传统的分枝定界算法做了初步的数值比较实验。计算实验表明算法对求解中大规模非凸二次规划问题的有效性。     

Earliness–tardiness minimization on a single machine to schedule preventive maintenance tasks: metaheuristic and exact methods

In this paper, we consider the problem of scheduling a set of M preventive maintenance tasks to be performed on M machines. The machines are assigned to execute production tasks. We aim to minimize the total preventive maintenance cost such that the maintenance tasks have to continuously be run during the schedule horizon. Such a constraint holds when the maintenance resources are not sufficient. We solve the problem by two exact methods and meta-heuristic algorithms. As exact procedures we used linear programming and branch and bound methods. As meta-heuristics, we propose a local search approach as well as a genetic algorithm. Computational experiments are performed on randomly generated instances to show that the proposed methods produce appropriate solutions for the problem. The computational results show that the deviation of the meta-heuristics solutions to the optimal one is very small, which confirms the effectiveness of meta-heuristics as new approaches for solving hard scheduling problems.     

Filtering Algorithms for the NValue Constraint

The NValue constraint counts the number of different values assigned to a vector of variables. Propagating generalized arc consistency on this constraint is NP-hard. We show that computing even the lower bound on the number of values is NP-hard. We therefore study different approximation heuristics for this problem. We introduce three new methods for computing a lower bound on the number of values. The first two are based on the maximum independent set problem and are incomparable to a previous approach based on intervals. The last method is a linear relaxation of the problem. This gives a tighter lower bound than all other methods, but at a greater asymptotic cost.     

Statistical properties of kernel principal component analysis

The main goal of this paper is to prove inequalities on the reconstruction error for kernel principal component analysis. With respect to previous work on this topic, our contribution is twofold: (1) we give bounds that explicitly take into account the empirical centering step in this algorithm, and (2) we show that a “localized” approach allows to obtain more accurate bounds. In particular, we show faster rates of convergence towards the minimum reconstruction error; more precisely, we prove that the convergence rate can typically be faster than n ^−1/2. We also obtain a new relative bound on the error. A secondary goal, for which we present similar contributions, is to obtain convergence bounds for the partial sums of the biggest or smallest eigenvalues of the kernel Gram matrix towards eigenvalues of the corresponding kernel operator. These quantities are naturally linked to the KPCA procedure; furthermore these results can have applications to the study of various other kernel algorithms. The results are presented in a functional analytic framework, which is suited to deal rigorously with reproducing kernel Hilbert spaces of infinite dimension. Editor: Nicolo Cesa-Bianchi An erratum to this article is available at .