Random backtracking in backtrack search algorithms for satisfiability

This paper proposes the utilization of randomized backtracking within complete backtrack search algorithms for propositional satisfiability (SAT). In recent years, randomization has become pervasive in SAT algorithms. Incomplete algorithms for SAT, for example the ones based on local search, often resort to randomization. Complete algorithms also resort to randomization. These include state-of-the-art backtrack search SAT algorithms that often randomize variable selection heuristics. Moreover, it is plain that the introduction of randomization in other components of backtrack search SAT algorithms can potentially yield new competitive search strategies. As a result, we propose a stochastic backtrack search algorithm for SAT, that randomizes both the variable selection and the backtrack steps of the algorithm. In addition, we relate randomized backtracking with a more general form of backtracking, referred to as unrestricted backtracking. Finally, experimental results for different organizations of randomized backtracking are described and compared, providing empirical evidence that the new search algorithm for SAT is a very competitive approach for solving hard real-world instances.     

Rollout Algorithms for Stochastic Scheduling Problems

Stochastic scheduling problems are difficult stochastic control problems with combinatorial decision spaces. In this paper we focus on a class of stochastic scheduling problems, the quiz problem and its variations. We discuss the use of heuristics for their solution, and we propose rollout algorithms based on these heuristics which approximate the stochastic dynamic programming algorithm. We show how the rollout algorithms can be implemented efficiently, with considerable savings in computation over optimal algorithms. We delineate circumstances under which the rollout algorithms are guaranteed to perform better than the heuristics on which they are based. We also show computational results which suggest that the performance of the rollout policies is near-optimal, and is substantially better than the performance of their underlying heuristics.     

A framework for the probabilistic analysis of hierarchical planning systems

As we have argued in previous papers, multi-level decision problems can often be modeled as multi-stage stochastic programs, and hierarchical planning systems designed for their solution, when viewed as stochastic programming heuristics, can be subjected to analytical performance evaluation. The present paper gives a general formulation of such stochastic programs and provides a framework for the design and analysis of heuristics for their solution. The various ways to measure the performance of such heuristics are reviewed, and some relations between these measures are derived. Our concepts are illustrated on a simple two-level planning problem of a general nature and on a more complicated two-level scheduling problem.     

二阶段随机规划问题基于随机模拟的遗传算法

利用遗传算法不过多依赖目标函数性质．适应于全局搜索的特点．提出了求解二阶段随机规划的基于随机模拟的遗传算法，算法采用随机模拟技术利用样本均值近似代替期望值，使计算得以简化，计算实例表明该算法是有效和可行的。     

A multi-heuristic approach for solving the pre-marshalling problem

Minimizing the number of reshuffling operations at maritime container terminals incorporates the pre-marshalling problem (PMP) as an important problem. Based on an analysis of existing solution approaches we develop new heuristics utilizing specific properties of problem instances of the PMP. We show that the heuristic performance is highly dependent on these properties. We introduce a new method that exploits a greedy heuristic of four stages, where for each of these stages several different heuristics may be applied. Instead of using randomization to improve the performance of the heuristic, we repetitively generate a number of solutions by using a combination of different heuristics for each stage. In doing so, only a small number of solutions is generated for which we intend that they do not have undesirable properties, contrary to the case when simple randomization is used. Our experiments show that such a deterministic algorithm significantly outperforms the original nondeterministic method. The improvement is twofold, both in the quality of found solutions, and in the computational effort.     

Random walk search procedures for reliability optimization of systems with fault tolerance

In this paper we develop two efficient discrete stochastic search methods based on random walk procedure for maximizing system reliability subjected to imperfect fault coverage where uncovered component failures cause immediate system failure, even in the presence of adequate redundancy. The first search method uses a sequential sampling procedure with fixed boundaries at each iteration. We show that this search process satisfies local balance equations and its equilibrium distribution gives most weight to the optimal solution. We also show that the solution that has been visited most often in the first m iterations converges almost surely to the optimal solution. The second search method uses a sequential sampling procedure with increasing boundaries at each iteration. We show that if the increase occurs slower than a certain rate, this search process will converge to the optimal set with probability 1. We consider the system where reliability cannot be evaluated exactly but must be estimated through Monte Carlo simulation. Copyright © 2008 John Wiley & Sons, Ltd.     

Search Heuristics for Resource Constrained Project Scheduling

We develop a search procedure for project scheduling problems with multiple resource constraints as well as precedence constraints. The procedure is applied to three popular search heuristics, simulated annealing, tabu search and genetic algorithms. In the heuristics, a solution is represented with a string of numbers each of which denotes priority of each activity. The priorities are used to select an activity for scheduling among competing ones. The search heuristics with this encoding method can always generate feasible neighbourhood solutions for a given solution. Moreover, this encoding method is very flexible in that problems with objective functions of a general functional form (such as a nonlinear function) and complex constraints can be considered without much difficulty. Results of computational tests on the performance of the search heuristics showed that the search heuristics, especially the simulated annealing and tabu search algorithms worked better than existing heuristics.     

A Stochastic Programming Model for Currency Option Hedging

In this paper we use a stochastic programming approach to develop currency option hedging models which can address problems with multiple random factors in an imperfect market. The portfolios considered in our model are rebalanced at the end of each time period, and reinvestments are allowed during the hedging process. These sequential decisions (reinvestments) are based on the evolution of random parameters such as exchange rates, interest rates, etc. We also allow the inclusion of a variety of instruments in the hedging portfolio, including short term derivative securities, short term options, and futures. These instruments help generate strategies that provide good liquidity and low trade intensity. One of the important features of the model is that it incorporates constraints on sensitivity measures such as Delta and Gamma. By ensuring that these hedge parameters track a desired trajectory (e.g., the parameters of a target option), the new model provides investment strategies that are robust with respect to the perturbations measured by Delta and Gamma. In order to manage the explosion of scenarios due to multiple random factors, we incorporate sampling within a scenario aggregation algorithm. We illustrate that when compared with other myopic hedging methods in imperfect markets, the new stochastic programming model can provide better performance. Our examples also illustrate stochastic programming as a practical computational tool for realistic hedging problems.     

All roads lead to Rome—New search methods for the optimal triangulation problem

To perform efficient inference in Bayesian networks by means of a Junction Tree method, the network graph needs to be triangulated. The quality of this triangulation largely determines the efficiency of the subsequent inference, but the triangulation problem is unfortunately NP-hard. It is common for existing methods to use the treewidth criterion for optimality of a triangulation. However, this criterion may lead to a somewhat harder inference problem than the total table size criterion. We therefore investigate new methods for depth-first search and best-first search for finding optimal total table size triangulations. The search methods are made faster by efficient dynamic maintenance of the cliques of a graph. This problem was investigated by Stix, and in this paper we derive a new simple method based on the Bron-Kerbosch algorithm that compares favourably to Stix’ approach. The new approach is generic in the sense that it can be used with other algorithms than just Bron-Kerbosch. The algorithms for finding optimal triangulations are mainly supposed to be off-line methods, but they may form the basis for efficient any-time heuristics. Furthermore, the methods make it possible to evaluate the quality of heuristics precisely and allow us to discover parts of the search space that are most important to direct randomized sampling to.     

Restart strategies for GRASP with path-relinking heuristics

GRASP with path-relinking is a hybrid metaheuristic, or stochastic local search (Monte Carlo) method, for combinatorial optimization. A restart strategy in GRASP with path-relinking heuristics is a set of iterations {i ₁, i ₂, …} on which the heuristic is restarted from scratch using a new seed for the random number generator. Restart strategies have been shown to speed up stochastic local search algorithms. In this paper, we propose a new restart strategy for GRASP with path-relinking heuristics. We illustrate the speedup obtained with our restart strategy on GRASP with path-relinking heuristics for the maximum cut problem, the maximum weighted satisfiability problem, and the private virtual circuit routing problem.     

Beam search heuristics for quadratic earliness and tardiness scheduling

In this paper, we present beam search heuristics for the single machine scheduling problem with quadratic earliness and tardiness costs, and no machine idle time. These heuristics include classic beam search procedures, as well as filtered and recovering algorithms. We consider three dispatching heuristics as evaluation functions, in order to analyse the effect of different rules on the performance of the beam search procedures. The computational results show that using better dispatching heuristics improves the effectiveness of the beam search algorithms. The performance of the several heuristics is similar for instances with low variability. For high variability instances, however, the detailed, filtered and recovering beam search (RBS) procedures clearly outperform the best existing heuristic. The detailed beam search algorithm performs quite well, and is recommended for small- to medium-sized instances. For larger instances, however, this procedure requires excessive computation times, and the RBS algorithm then becomes the heuristic of choice.     

Dynamic uncapacitated lot sizing with random demand under a fillrate constraint

This paper deals with the single-item dynamic uncapacitated lot sizing problem with random demand. We propose a model based on the “static uncertainty” strategy of Bookbinder and Tan (1988). In contrast to these authors, we use exact expressions for the inventory costs and we apply a fillrate constraint. We present an exact solution method and modify several well-known dynamic lot sizing heuristics such that they can be applied for the case of dynamic stochastic demands. A numerical experiment shows that there are significant differences in the performance of the heuristics whereat the ranking of the heuristics is different from that reported for the case of deterministic demand.     

Johnson’s problem with stochastic processing times and optimal service level

Theoretical results about Johnson’s problem with stochastic processing times are few. In general, just finding the expected makespan of a given sequence is already difficult, even for discrete processing time distributions. Furthermore, to obtain optimal service level we need to compute the entire distribution of the makespan. Therefore the use of heuristics and simulation is justified. We show that pursuing the minimal expected makespan by two heuristics is empirically effective for obtaining excellent overall distributions. The first is to use Johnson’s rule on the means. The second is based on pair-switching and converges to some known stochastically optimal solutions when they apply. We show that the first heuristic is asymptotically optimal under mild conditions. We also investigate the effect of sequencing on the makespan variance.     

Solving the Vehicle Routing Problem with Stochastic Demands using the Cross-Entropy Method

An alternate formulation of the classical vehicle routing problem with stochastic demands (VRPSD) is considered. We propose a new heuristic method to solve the problem, based on the Cross-Entropy method. In order to better estimate the objective function at each point in the domain, we incorporate Monte Carlo sampling. This creates many practical issues, especially the decision as to when to draw new samples and how many samples to use. We also develop a framework for obtaining exact solutions and tight lower bounds for the problem under various conditions, which include specific families of demand distributions. This is used to assess the performance of the algorithm. Finally, numerical results are presented for various problem instances to illustrate the ideas.     

Negative Effects of Modeling Techniques on Search Performance

Symmetries occur in many combinatorial problems, and a great deal of research has been done on symmetry breaking techniques for backtrack search. However, few results have been reported on the use of symmetry breaking with local search. On four classes of problem we find that adding symmetry breaking constraints to a model impairs local search performance, in terms of both execution time and search steps. We also find that implied constraints can impair backtrack search performance. These results show that modeling techniques and search heuristics should be combined with caution. They also motivate a novel modeling technique for local search: removing constraints to add new symmetries.     

Investigation of practical,robust and flexible decisions for facility location problems using tabu search and simulation

We investigate how robust and flexible solutions of stochastic capacitated facility location problems (CFLPs) can be obtained by combining metaheuristic optimization with Monte Carlo sampling techniques. To this end, we develop a tabu search procedure for the CFLP, and use this to solve an extensive set of stochastic versions of this problem.     

Time-inhomogeneous discrete stochastic search methods for optimal Bernoulli parameters

We present two time-inhomogeneous search processes for finding the optimal Bernoulli parameters, where the performance measure cannot be evaluated exactly but must be estimated through Monte Carlo simulation. At each iteration, two neighbouring alternatives are compared and the one that appears to be better is passed on to the next iteration. The first search process uses an increasing sample size of each configuration at each iteration. The second search process uses a sequential sampling procedure with increasing boundaries as the number of iterations increases. At each iteration the acceptance of a new configuration depends on the iterate number, therefore, the search process turns out to be inhomogeneous Markov chain. We show that if the increase occurs slower than a certain rate, these search processes will converge to the optimal set with probability one. © 1998 John Wiley & Sons, Ltd.     

Performance of a genetic algorithm for the graph partitioning problem

The performance of the genetic algorithm (GA) for the graph partitioning problem (GPP) is investigated by comparison with standard heuristics on well-known benchmark graphs. In general, there is a case where a practical performance of a conventional genetic approach, which performs only simple operations without a local search strategy, is not sufficient. However, it is known that a combination of GA and local search can produce better solutions. From this practice, we incorporate a simple local search algorithm into the GA. In particular, the search ability of the GA is compared with standard heuristics such as multistart local search and simulated annealing, which use the same neighborhood structure of the simple local search, for solving the GPP. Experimental results show that the GA performs better than its competitors.     

Uniform exponential convergence of sample average random functions under general sampling with applications in stochastic programming

Sample average approximation (SAA) is one of the most popular methods for solving stochastic optimization and equilibrium problems. Research on SAA has been mostly focused on the case when sampling is independent and identically distributed (iid) with exceptions (Dai et al. (2000) [9], Homem-de-Mello (2008) [16]). In this paper we study SAA with general sampling (including iid sampling and non-iid sampling) for solving nonsmooth stochastic optimization problems, stochastic Nash equilibrium problems and stochastic generalized equations. To this end, we first derive the uniform exponential convergence of the sample average of a class of lower semicontinuous random functions and then apply it to a nonsmooth stochastic minimization problem. Exponential convergence of estimators of both optimal solutions and M-stationary points (characterized by Mordukhovich limiting subgradients (Mordukhovich (2006) [23], Rockafellar and Wets (1998) [32])) are established under mild conditions. We also use the unform convergence result to establish the exponential rate of convergence of statistical estimators of a stochastic Nash equilibrium problem and estimators of the solutions to a stochastic generalized equation problem.