Generalised graph colouring by a hybrid of local search and constraint programming

The IMPASSE class of local search algorithms have given good results on many vertex colouring benchmarks. Previous work enhanced IMPASSE by adding the constraint programming technique of forward checking, in order to prune colouration neighbourhoods during search. On several large graphs the algorithm found the best known colourings. This paper extends the work by improving the heuristics and generalising the approach to bandwidth multicolouring. It is shown to give better results than a related search algorithm on an integer programming model, and to be competitive with published results. Experiments indicate that stronger constraint propagation further improves search performance, but that a symmetry breaking technique has unpredictable effects.     

Local search and constraint programming for the post enrolment-based course timetabling problem

We present a variety of approaches for solving the post enrolment-based course timetabling problem, which was proposed as Track 2 of the 2007 International Timetabling Competition. We approach the problem using local search and constraint programming techniques. We show how to take advantage of a list-colouring relaxation of the problem. Our local search approach won Track 2 of the 2007 competition. Our best constraint programming approach uses an original problem decomposition. Incorporating this into a large neighbourhood search scheme seems promising, and provides motivation for studying complete approaches in further detail.     

Constrained global optimization of multivariate polynomials using Bernstein branch and prune algorithm

We propose an algorithm for constrained global optimization to tackle non-convex nonlinear multivariate polynomial programming problems. The proposed Bernstein branch and prune algorithm is based on the Bernstein polynomial approach. We introduce several new features in this proposed algorithm to make the algorithm more efficient. We first present the Bernstein box consistency and Bernstein hull consistency algorithms to prune the search regions. We then give Bernstein contraction algorithm to avoid the computation of Bernstein coefficients after the pruning operation. We also include a new Bernstein cut-off test based on the vertex property of the Bernstein coefficients. The performance of the proposed algorithm is numerically tested on 13 benchmark problems. The results of the tests show the proposed algorithm to be overall considerably superior to existing method in terms of the chosen performance metrics.     

Accelerating Branch-and-Bound through a Modeling Language Construct for Relaxation-Specific Constraints

In the tradition of modeling languages for optimization, a single model is passed to a solver for solution. In this paper, we extend BARON’s modeling language in order to facilitate the communication of problem-specific relaxation information from the modeler to the branch-and-bound solver. This effectively results into two models being passed from the modeling language to the solver. Three important application areas are identified and computational experiments are presented. In all cases, nonlinear constraints are provided only to the relaxation constructor in order to strengthen the lower bounding step of the algorithm without complicating the local search process. In the first application area, nonlinear constraints from the reformulation–linearization technique (RLT) are added to strengthen a problem formulation. This approach is illustrated for the pooling problem and computational results show that it results in a scheme that makes global optimization nearly as fast as local optimization for pooling problems from the literature. In the second application area, we communicate with the relaxation constructor the first-order optimality conditions for unconstrained global optimization problems. Computational experiments with polynomial programs demonstrate that this approach leads to a significant reduction of the size of the branch-and-bound search tree. In the third application, problem-specific nonlinear optimality conditions for the satisfiability problem are used to strengthen the lower bounding step and are found to significantly expedite the branch-and-bound algorithm when applied to a nonlinear formulation of this problem.     

Combining the Scalability of Local Search with the Pruning Techniques of Systematic Search

Systematic backtracking is used in many constraint solvers and combinatorial optimisation algorithms. It is complete and can be combined with powerful search pruning techniques such as branch-and-bound, constraint propagation and dynamic variable ordering. However, it often scales poorly to large problems. Local search is incomplete, and has the additional drawback that it cannot exploit pruning techniques, making it uncompetitive on some problems. Nevertheless its scalability makes it superior for many large applications. This paper describes a hybrid approach called Incomplete Dynamic Backtracking, a very flexible form of backtracking that sacrifices completeness to achieve the scalability of local search. It is combined with forward checking and dynamic variable ordering, and evaluated on three combinatorial problems: on the n-queens problem it out-performs the best local search algorithms; it finds large optimal Golomb rulers much more quickly than a constraint-based backtracker, and better rulers than a genetic algorithm; and on benchmark graphs it finds larger cliques than almost all other tested algorithms. We argue that this form of backtracking is actually local search in a space of consistent partial assignments, offering a generic way of combining standard pruning techniques with local search.     

Analysis and Solving SAT and MAX-SAT Problems Using an L-partition Approach

In this paper, we study SAT and MAX-SAT using the integer linear programming models and L-partition approach. This approach can be applied to analyze and solve many discrete optimization problems including location, covering, scheduling problems. We describe examples of SAT and MAX-SAT families for which the cardinality of L-covering of the relaxation polytope grows exponentially with the number of variables. These properties are useful in analysis and development of algorithms based on the linear relaxation of the problems. Besides we present the L-class enumeration algorithm for SAT using the L-partition approach. In addition we consider an application of this algorithm to construct exact algorithm and local search algorithms for the MAX-SAT problem.     

A multiple-tree search procedure for the resource-constrained project scheduling problem

For over three decades, researchers have sought effective solution procedures for PERT/CPM types of scheduling problems under conditions of limited resource availability. This paper presents a procedure for this problem which takes advantage of the emerging technology provided by multiple parallel processors to find and verify an optimal schedule for a project under conditions of multiple resource constraints. In our approach, multiple solutions trees are searched simultaneously in the quest for a minimum duration schedule. Global upper and lower bound information in common memory is shared among processors, enabling one or several processors to prune potentially significant portions of its search tree based upon bounds discovered by a processor using a different search tree. Computational experience is reported both for problems in which resources are available in constant amounts per period, as well as the much more difficult problem in which the resources available are allowed to vary over the schedule horizon (e.g., travel, sick leave, assignment to other tasks or projects, and so forth). The modular multiple-tree search procedure described in this paper is quite general, permitting most types of existing serial search strategies to be adapted to this approach where multiple processors are available.     

Impulsively-controlled systems and reverse dwell time: A linear programming approach

We present a receding horizon algorithm that converges to the exact solution in polynomial time for a class of optimal impulse control problems with uniformly distributed impulse instants and governed by so-called reverse dwell time conditions. The cost has two separate terms, one depending on time and the second monotonically decreasing on the state norm. The obtained results have both theoretical and practical relevance. From a theoretical perspective we prove certain geometrical properties of the discrete set of feasible solutions. From a practical standpoint, such properties reduce the computational burden and speed up the search for the optimum thus making the algorithm suitable for the on-line implementation in real-time problems. Our approach consists in approximating the optimal impulse control problem via a binary linear programming problem with a totally unimodular constraint matrix. Hence, solving the binary linear programming problem is equivalent to solving its linear relaxation. Then, given the feasible solution from the linear relaxation, we find the optimal solution via receding horizon and local search. Numerical illustrations of a queueing system are performed.     

An exact method with variable fixing for solving the generalized assignment problem

We propose a simple exact algorithm for solving the generalized assignment problem. Our contribution is twofold: we reformulate the optimization problem into a sequence of decision problems, and we apply variable-fixing rules to solve these effectively. The decision problems are solved by a simple depth-first lagrangian branch-and-bound method, improved by our variable-fixing rules to prune the search tree. These rules rely on lagrangian reduced costs which we compute using an existing but little-known dynamic programming algorithm.     

An efficient computational method for a stochastic dynamic lot-sizing problem under service-level constraints

We provide an efficient computational approach to solve the mixed integer programming (MIP) model developed by Tarim and Kingsman [8] for solving a stochastic lot-sizing problem with service level constraints under the static-dynamic uncertainty strategy. The effectiveness of the proposed method hinges on three novelties: (i) the proposed relaxation is computationally efficient and provides an optimal solution most of the time, (ii) if the relaxation produces an infeasible solution, then this solution yields a tight lower bound for the optimal cost, and (iii) it can be modified easily to obtain a feasible solution, which yields an upper bound. In case of infeasibility, the relaxation approach is implemented at each node of the search tree in a branch-and-bound procedure to efficiently search for an optimal solution. Extensive numerical tests show that our method dominates the MIP solution approach and can handle real-life size problems in trivial time.     

A Seeded Memetic Algorithm for Large Unit Commitment Problems

The paper shows that the use of a memetic algorithm (MA), a genetic algorithm (GA) combined with local search, synergistically combined with Lagrangian relaxation is effective and efficient for solving large unit commitment problems in electric power systems. It is shown that standard implementations of GA or MA are not competitive with the traditional methods of dynamic programming (DP) and Lagrangian relaxation (LR). However, an MA seeded with LR proves to be superior to all alternatives on large problems. Eight problems from the literature and a new large, randomly generated problem are used to compare the performance of the proposed seeded MA with GA, MA, DP and LR. Compared with previously published results, this hybrid approach solves the larger problems better and uses less computational time.     

Intelligent variable orderings and re-orderings in DAC-based solvers for WCSP

Weighted constraint satisfaction problems (WCSPs) is a well-known framework for combinatorial optimization problems with several domains of application. In the last few years, several local consistencies for WCSPs have been proposed. Their main use is to embed them into a systematic search, in order to detect and prune unfeasible values as well as to anticipate the detection of deadends. Some of these consistencies rely on an order among variables but nothing is known about which orders are best. Therefore, current implementations use the lexicographic order by default. In this paper we analyze the effect of heuristic orders at three levels of increasing overhead: i) compute the order prior to search and keep it fixed during the whole solving process (we call this a static order), ii) compute the order at every search node using current subproblem information (we call this a dynamic order) and iii) compute a sequence of different orders at every search node and sequentially enforce the local consistency for each one (we call this dynamic re-ordering). We performed experiments in three different problems: Max-SAT, Max-CSP and warehouse location problems. We did not find an alternative better than the rest for all the instances. However, we found that inverse degree (static order), sum of unary weights (dynamic order) and re-ordering with the sum of unary weights are good heuristics which are always better than a random order. This research is supported by the MEC through project TIC-2002-04470-C03.     

On some convexity properties of the Least Squares Method for pairwise comparisons matrices without the reciprocity condition

The relaxation of the reciprocity condition for pairwise comparisons is revisited from the optimization point of view. We show that some special but not extreme cases of the Least Squares Method are easy to solve as convex optimization problems after suitable nonlinear change of variables. We also give some other, less restrictive conditions under which the convexity of a modified problem can be assured, and the global optimal solution of the original problem found by using local search methods. Mathematical and psychological justifications for the relaxation of the reciprocity condition as well as numerical examples are provided.     

Transition Manifolds of Complex Metastable Systems

We consider complex dynamical systems showing metastable behavior, but no local separation of fast and slow time scales. The article raises the question of whether such systems exhibit a low-dimensional manifold supporting its effective dynamics. For answering this question, we aim at finding nonlinear coordinates, called reaction coordinates, such that the projection of the dynamics onto these coordinates preserves the dominant time scales of the dynamics. We show that, based on a specific reducibility property, the existence of good low-dimensional reaction coordinates preserving the dominant time scales is guaranteed. Based on this theoretical framework, we develop and test a novel numerical approach for computing good reaction coordinates. The proposed algorithmic approach is fully local and thus not prone to the curse of dimension with respect to the state space of the dynamics. Hence, it is a promising method for data-based model reduction of complex dynamical systems such as molecular dynamics.     

Two phased hybrid local search for the periodic capacitated arc routing problem

The periodic capacitated arc routing problem (PCARP) is a challenging general model with important applications. The PCARP has two hierarchical optimization objectives: a primary objective of minimizing the number of vehicles (F_v) and a secondary objective of minimizing the total cost (F_c). In this paper, we propose an effective two phased hybrid local search (HLS) algorithm for the PCARP. The first phase makes a particular effort to optimize the primary objective while the second phase seeks to further optimize both objectives by using the resulting number of vehicles of the first phase as an upper bound to prune the search space. For both phases, combined local search heuristics are devised to ensure an effective exploration of the search space. Experimental results on 63 benchmark instances demonstrate that HLS performs remarkably well both in terms of computational efficiency and solution quality. In particular, HLS discovers 44 improved best known values (new upper bounds) for the total cost objective F_c while attaining all the known optimal values regarding the objective of the number of vehicles F_v. To our knowledge, this is the first PCARP algorithm reaching such a performance. Key components of HLS are analyzed to better understand their contributions to the overall performance.     

A hybrid optimization technique coupling an evolutionary and a local search algorithm

Evolutionary algorithms are robust and powerful global optimization techniques for solving large-scale problems that have many local optima. However, they require high CPU times, and they are very poor in terms of convergence performance. On the other hand, local search algorithms can converge in a few iterations but lack a global perspective. The combination of global and local search procedures should offer the advantages of both optimization methods while offsetting their disadvantages. This paper proposes a new hybrid optimization technique that merges a genetic algorithm with a local search strategy based on the interior point method. The efficiency of this hybrid approach is demonstrated by solving a constrained multi-objective mathematical test-case.     

A variable neighborhood search with an effective local search for uncapacitated multilevel lot-sizing problems

In this study, we improved the variable neighborhood search (VNS) algorithm for solving uncapacitated multilevel lot-sizing (MLLS) problems. The improvement is twofold. First, we developed an effective local search method known as the Ancestors Depth-first Traversal Search (ADTS), which can be embedded in the VNS to significantly improve the solution quality. Second, we proposed a common and efficient approach for the rapid calculation of the cost change for the VNS and other generate-and-test algorithms. The new VNS algorithm was tested against 176 benchmark problems of different scales (small, medium, and large). The experimental results show that the new VNS algorithm outperforms all of the existing algorithms in the literature for solving uncapacitated MLLS problems because it was able to find all optimal solutions (100%) for 96 small-sized problems and new best-known solutions for 5 of 40 medium-sized problems and for 30 of 40 large-sized problems.     

Progress control in iterated local search for nurse rostering

This paper describes an approach in which a local search technique is alternated with a process which ‘jumps’ to another point in the search space. After each ‘jump’ a (time-intensive) local search is used to obtain a new local optimum. The focus of the paper is in monitoring the progress of this technique on a set of real world nurse rostering problems. We propose a model for estimating the quality of this new local optimum. We can then decide whether to end the local search based on the predicted quality. The fact that we avoid searching these bad neighbourhoods enables us to reach better solutions in the same amount of time. We evaluate the approach on five highly constrained problems in nurse rostering. These problems represent complex and challenging real world rostering situations and the approach described here has been developed during a commercial implementation project by ORTEC bv.     

Exploring relaxation induced neighborhoods to improve MIP solutions

Given a feasible solution to a Mixed Integer Programming (MIP) model, a natural question is whether that solution can be improved using local search techniques. Local search has been applied very successfully in a variety of other combinatorial optimization domains. Unfortunately, local search relies extensively on the notion of a solution neighborhood, and this neighborhood is almost always tailored to the structure of the particular problem being solved. A MIP model typically conveys little information about the underlying problem structure. This paper considers two new approaches to exploring interesting, domain-independent neighborhoods in MIP. The more effective of the two, which we call Relaxation Induced Neighborhood Search (RINS), constructs a promising neighborhood using information contained in the continuous relaxation of the MIP model. Neighborhood exploration is then formulated as a MIP model itself and solved recursively. The second, which we call guided dives, is a simple modification of the MIP tree traversal order. Loosely speaking, it guides the search towards nodes that are close neighbors of the best known feasible solution. Extensive computational experiments on very difficult MIP models show that both approaches outperform default CPLEX MIP and a previously described approach for exploring MIP neighborhoods (local branching) with respect to several different metrics. The metrics we consider are quality of the best integer solution produced within a time limit, ability to improve a given integer solution (of both good and poor quality), and time required to diversify the search in order to find a new solution.Mathematics Subject Classification (2000):20E28, 20G40, 20C20Acknowledgement We wish to thank the two anonymous referees for their helpful comments.     

SPT: a stochastic tunneling algorithm for global optimization

A stochastic approach to solving unconstrained continuous-function global optimization problems is presented. It builds on the tunneling approach to deterministic optimization presented by Barhen and co-workers (Bahren and Protopopescu, in: State of the Art in Global Optimization, Kluwer, 1996; Barhen et al., Floudas and Pardalos (eds.), TRUST: a deterministic algorithm for global optimization, 1997) by combining a series of local descents with stochastic searches. The method uses a rejection-based stochastic procedure to locate new local minima descent regions and a fixed Lipschitz-like constant to reject unpromising regions in the search space, thereby increasing the efficiency of the tunneling process. The algorithm is easily implemented in low-dimensional problems and scales easily to large problems. It is less effective without further heuristics in these latter cases, however. Several improvements to the basic algorithm which make use of approximate estimates of the algorithms parameters for implementation in high-dimensional problems are also discussed. Benchmark results are presented, which show that the algorithm is competitive with the best previously reported global optimization techniques. A successful application of the approach to a large-scale seismology problem of substantial computational complexity using a low-dimensional approximation scheme is also reported.