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Greedy Randomized Adaptive Path Relinking Silvio Binato∗
Haroldo Faria Jr.∗
Mauricio G. C. Resende†
∗
Brazilian Electric Energy Research Center – CEPEL P.O.Box 68007, Rio de Janeiro, RJ 21944-970 Brazil. Email: {silvio, hfjr}@cepel.br
†
1
Information Sciences Research Center, AT&T Labs–Research Florham Park, NJ 07932 USA. Email:
[email protected]
Introduction
A wide spectrum of “real world” problems, such as vehicle routing, machine and crew scheduling, facility location, portfolio analysis, electricity generation planning, and communication and transportation network design, demands the use of combinatorial optimization methods. By a combinatorial optimization problem, we mean a program where a linear or nonlinear objective function must be optimized satisfying a finite set of (linear or nonlinear) constraints where some, or all, variables are required to be integer. Different optimization techniques have been proposed to deal with combinatorial problems. First, there are the class of formal mathematical methods, which are characterized by their ability to obtain the optimal solution. Examples include cutting planes, branch-and-bound, and branch-and-cut. Nevertheless, all formal mathematical approaches are non-polynomial time methods, i.e. they may require exponential time to find the optimal solution, and hence they become impractical as the sizes of the problem instances grow. A feasible alternative to formal mathematical algorithms is heuristic approaches (antique Greek word “heuiskein” whose meaning is to discover, to invent, to have an idea). These kind of methods are unable to prove the optimality of the solution found, but can provide high quality solutions in an acceptable computation time, even for large-scale and complex problems. In the last years, since the introduction of metaheuristics (heuristics for managing heuristics), significant developments have been made in this area. The best known metaheuristics include Tabu Search [7, 8], Simulating Annealing [13], Genetic Algorithms [11, 12], Scatter Search [9] and GRASP [5]. Regarding GRASP, a quick bibliography review allows the enumerations of a very large number of articles and technical reports illustrating successful applications to “real world” combinatorial problems, see [6] for an annotated bibliography of GRASP. This large number of applications of GRASP to solve combinatorial problems is explained by its simplicity, power, and ease of implementation. The term Path Relinking was coined by Glover et al., see [10], to denominate a new search procedure to be applied in the solution of combinatorial problems. Contrary to GRASP, this approach has not yet been applied to a large number of problems. However, the studies realized so far have indicated that Path Relinking is very effective for improving the quality of solutions in combinatorial optimization problems. Laguna and Mart´ı [14] introduced path relinking within GRASP as a form of intensification. See [1] for refinements of path relinking for GRASP. Porto, Portugal, July 16-20, 2001
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MIC’2001 - 4th Metaheuristics International Conference procedure GRASP(ListSize, M axIter, RandomSeed) 1 2 3 4 5 6
for k = 1, . . . , M axIter,do; ConstructSolution(ListSize, RandomSeed); LocalSearch(BestSolutionF ound); UpdateSolution(BestSolutionF ound); rof; return BestSolutionF ound
end GRASP; Figure 1: A generic GRASP pseudo-code.
In this paper we present a new search procedure that combines GRASP concepts and those of Path Relinking. Summarizing, original Path Relinking finds a path between two “good” solutions in order to discover new ones, potentially better than the older solutions. GRASP’s basic mechanisms are the greedy randomized construction phase, where a feasible solution is built, and the local search procedure, where the neighborhood of the solution obtained is explored. Greedy randomized adaptive path relinking (GRAPR) constructs a GRASP to build different paths in a Path Relinking phase. This paper is organized as follows. The next section makes a brief review of GRASP. Section 3 presents the basic concepts of Path Relinking. Section 4 discusses the new approach, greedy randomized adaptive path relinking, and presents some examples how it works in combinatorial search methods. Finally, Section 5 presents conclusions and discusses research paths for future investigation.
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A Brief Review of GRASP
GRASP [4, 5] is an iterative process, see Figure 1 for an illustration of a generic pseudo-code of a GRASP implementation, where each GRASP iteration consists of two phases: construction (line 2), and local search (line 3). The best solution over all GRASP iterations is returned as the result (lines 4 and 6). In the construction phase, a feasible solution is built, one element at a time. At each construction iteration, the next element to be added is determined by ordering all elements in a candidate list with respect to a greedy function that measures the near-term benefit of selecting each element. This list is called the restricted candidate list (RCL). The adaptive component of the heuristic arises from the fact that the benefits associated with every element are updated at each iteration of the construction phase to reflect the changes brought on by the selection of the previous elements. The probabilistic component of a GRASP is characterized by randomly choosing one of the best candidates in the RCL, but usually not the best one. This way of making the choice allows for different solutions to be obtained at each GRASP iteration, while not necessarily jeopardizing the adaptive greedy component. The solutions generated by a GRASP construction phase are not guaranteed to be locally optimal with respect to simple neighborhood definitions. Hence, it is almost always beneficial to apply a local search to attempt to improve each constructed solution. A local search algorithm works in an iterative fashion by successively replacing the current solution by a better solution from its neighborhood. It terminates when there is no better solution found in the neighborhood with respect to some cost function. Input for GRASP includes parameters for setting the candidate list size and the maximum number of GRASP iterations, and the seed for the random number generator. Porto, Portugal, July 16-20, 2001
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Figure 2: Path generation by a Path Relinking
3
Path Relinking
Path Relinking is an evolutionary approach that has been used to solve combinatorial optimization problems from both classical and real-world settings. It is a generalized form of the scatter search approach and is related to the tabu search metaheuristic since it attempts to integrate intensification and diversification strategies. Path Relinking works on a population of (good) solutions combining them to obtain new ones. These new solutions are generated by exploring trajectories that connect high-quality solutions and has this name because solutions are linked by a series of moves during the search and characteristically relinks previous points in ways not achieved in previous search history. In order to generate these paths, one must select a source and a guide solution. It is also possible to have a set of guiding solutions (multiple parents) generating combinations of elite solutions linking points in the space in various ways. In the process of linking a solution A (initial reference point) to a solution B (desired or guiding point), see Figure 2, a path is constructed by greedy move selections with respect to the objective function evaluation, that is, a neighborhood move is selected if it minimizes the value of the objective function in a local sense. During path relinking, the main objective is to incorporate attributes of the guiding solution (or solutions) while recording values of the objective function. The purpose of these relinking moves is to obtain improved solutions in the neighborhood of the solutions visited by original paths. Through an iterative procedure, the trajectory from A to B is generated by selecting the greedy Aneighbor solution (we will call this solution by C), from a set of all neighbors which decrease the distance from A to B. This distance function may be evaluated by the number of differences between both solutions (A and B). This procedure is restarted, making A ← C, until the target solution B is obtained. A variant of Path Relinking can also start with both endpoints simultaneously producing two sequences that must have identical final values.
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Greedy randomized adaptive path relinking
The main objective of greedy randomized adaptive path relinking (GRAPR) is to make a deep exploration of paths between the initial reference and the guiding solutions (the path’s endpoints). Note that may exist a very large number of different paths between the two endpoints. Each path Porto, Portugal, July 16-20, 2001
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Figure 3: Paths generated in the construction phase of GRASP Path Relinking
Figure 4: Bundle of trajectories considered by one iteration of the GRASP Path Relinking
explores a different part of the search space and, of course, may contain potentially good solutions. However, the traditional Path Relinking generates only one or two path between the endpoints, from the initial reference solution to the guiding solution, or vice-versa. This path, as illustrated in section 3, is generated using greedy movements regarding the objective function of the problem. The idea of GRAPR is to apply GRASP concepts in order to randomize the movements which characterize the Path Relinking trajectory. To implement this, we need to generalize both the construction and the local search phases of GRASP to the trajectory selection. Using the same example of section 3, in the construction phase we will build not only one path between A and B, but many paths. To generate these paths, the A-neighbor is selected, at random, from a candidate list composed by the best A-neighbors, ranked by the objective function. This phase differs from the original Path Relinking because it uses a random selection to select a neighbor, instead of greedy selection. Therefore, the trajectories generated by the construction phase will make an better exploitation of the search space, introducing a degree of diversification. The Figure 3 illustrates 4 paths connecting solutions A to B. Two possibilities can be explored to implement the local search phase for this new method. First, we can apply the traditional GRASP local search in a group of (or in all) solutions of the trajectory generated in the construction phase. Second, we can explore the neighborhood of the trajectory generated by the construction phase. This means that a bundle of paths will be considered each iteration of GRAPR, as illustrated in Figure 4. This last possibility makes more sense regarding to GRAPR, but requires more computational resources. Porto, Portugal, July 16-20, 2001
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Conclusions
In this extended abstract we introduced a new search procedure to solve combinatorial optimization problems called greedy randomized adaptive path relinking (GRAPR). This method consists of the generalization of GRASP concepts (construction and local search phases) to the Path Relinking approach. We have seen that GRAPR will produce a better exploitation of search space around potentially “good” solutions. We have also seen that, due to its characteristics, GRAPR can be an alternative approach to substitute the local search phase of traditional GRASP. An implementation of GRAPR is under development to a set of traditional combinatorial problems (e.g. job-shop scheduling [2], three-index assignment problem [1], and power transmission network design problems [3]. Computational results will be included in the final version of this paper.
References [1] R.M. Aiex, M.G.C. Resende, P.M. Pardalos, and G. Toraldo. GRASP with path relinking for the three-index assignment problem. Technical report, AT&T Labs Research, Florham Park, NJ 07733, 2000. [2] S. Binato, W.J. Hery, D. Loewenstern, and M.G.C. Resende. A greedy randomized adaptive search procedure for job shop scheduling. In C.C. Ribeiro and P. Hansen, editors, Essays and surveys on metaheuristics. Kluwer Academic Publishers, 2001. [3] S. Binato and G.C. Oliveira. A reactive GRASP for transmission network expansion planning. In C.C. Ribeiro and P. Hansen, editors, Essays and surveys on metaheuristics. Kluwer Academic Publishers, 2001. [4] T.A. Feo and M.G.C. Resende. A probabilistic heuristic for a computationally difficult set covering problem. Operations Research Letters, 8:67–71, 1989. [5] T.A. Feo and M.G.C. Resende. Greedy randomized adaptive search procedures. J. of Global Optimization, 6:109–133, 1995. [6] P. Festa and M.G.C. Resende. GRASP: An annotated bibliography. In C.C. Ribeiro and P. Hansen, editors, Essays and surveys on metaheuristics. Kluwer Academic Publishers, 2001. [7] F. Glover. Tabu Seach - part I. ORSA Journal on Computing, 1:90–206, 1989. [8] F. Glover. Tabu Seach - part II. ORSA Journal on Computing, 2:4–32, 1990. [9] F. Glover. Genetic algorithms and scatter search: Unsuspected potentials. Statistics and Computing, 4:131–140, 1994. [10] F. Glover, M. Laguna, and R. Mart´ı. Fundamentals of scatter search and path relinking. Control and Cybernetics, 39:653–684, 2000. [11] D. Goldberg. Genetic Algorithms in Search, Optimization and Machine Learning. Addison-Wesley, 1989. [12] J. Holland. Adaptation in Natural and Artificial System. University of Michigan Press, 1975. [13] S. Kirkpatrick. Optimization by simulating annealing: quantitative studies. Journal of Statistical Physics, 34:975–986, 1984. [14] M. Laguna and R. Mart´ı. GRASP and path relinking for 2-layer straight line crossing minimization. INFORMS J. on Computing, 11:44–52, 1999.
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