MIC2001 - 4th Metaheuristics International Conference 333

Finite Convergence of Tabu Search

Fred Glover Sad Hana

Hearin Center for Enterprise ScienceSchool of Business Administration

University of MississippiUniversity, MS 38677 USA

Email: fglover@bus.olemiss.edu

LAMIH - UMR CNRS n 8530Unite de Recherche Operationnelle et dAide a` la Decision

Universite de Valenciennes et du Hainaut-CambresisLe Mont Houy - B.P. 311 - 59304 Valenciennes Cedex - France

Email: Said.Hanafi@univ-valenciennes.fr

1 Introduction

Many optimization techniques (both heuristic and exact) for solving combinatorial and nonlinear prob-lems are iterative neighborhood search procedures i.e., they start with an initial solution (feasible orinfeasible) and repeatedly construct new solutions from current solutions by moves dened by referenceto a neighborhood structure. The process continues to generate a trajectory of neighboring solutionsuntil a certain stopping criterion is satised.

The adaptive memory approach of Tabu Search (TS) generates a neighborhood trajectory by includinga mechanism that forbids the search to revisit solutions already encountered unless the interveningtrajectory is modied (see [7]). The main goal of memory structures in TS is not simply to forbidcycling, however. In fact, the choice of a given neighborhood and a decision criterion for selectingmoves with TS can force some solutions to be revisited before exploring other new ones. Within thiscontext, a proposal of Glover [4] identies a simple rule for revisiting solutions that is conjectured tohave implications for niteness in zero-one integer programming and optimal set membership problems.Hana [8] proves Glovers conjecture under the assumption that the graph of the neighborhood spaceis connected and symmetric.

In this paper, we provide new proofs that yield specic bounds establishing the nite convergence oftabu search, specically for certain TS algorithms based on recency memory or frequency memory. Ourresults distinguish between symmetric and asymmetric neighborhood structures and provide insightsinto the sequences of solutions generated by the search. The outcomes disclose interesting contrastsbetween TS trajectories and the those generated by the more rigid rules underlying tree search methods.Based on these ndings, we also give designs for more ecient forms of convergent tabu search, andprovide special rules that create a new type of tree search.

This work is the rst to provide explicit convergence bounds for methods based on such forms ofmemory. The niteness of these methods suggests an important distinction between their underlyingideas and the rationale that gives rise to innite time convergence results for certain randomizedprocedures such as annealing.

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334 MIC2001 - 4th Metaheuristics International Conference

2 A Convergent Tabu Search

Let T ime(x) = the most recent time (iteration) that solution x was visited by a search process, whoseform is determined as follows.

Convergent Tabu Search (CTS)

Step 0 Initialization, the values T ime(x), x X , begin as arbitrary nonnegative integers, and the startingsolution x for the search is assigned a value so that T ime(x) > Time(x) for all x other thanx. (This includes the case where we begin with T ime(x) = 0 for all x X except x. Select astarting solution x X and set Iteration = 0;

Step 1 Set T ime(x) = Iteration. Select an unvisited neighbor x N(x) such that T ime(x) = 0, if oneexists, and otherwise choose to visit a solution x = argmin{T ime(y) : y N(x)}; T ime(x) =Iteration.

Step 2 Stop if all solutions in X are visited i.e., T ime(x) > 0 x X . Otherwise set Iteration =Iteration+ 1; x = x (move from x to x); go to Step 1.

Theorem ADenote the cardinality of X by n = |X |, and consider a value Un for n 2 which is given recursively byU1 = 0 and dene Un+1 = 2Un + 1, for n 1. Beginning with any solution x X , the CTS methodwill visit every solution in X in at most Un steps if X is nite and there exists a neighborhood pathfrom every solution in X to every other solution in X .

The min{T ime(x)} rule is the one called the Aspiration by Default rule in the TS literature.Since frequency-based memory is also useful in TS, it is natural to speculate that a frequency versionof Theorem A is valid. We apply the natural denition, Frequency(x) = the number of times x basbeen visited, and replace T ime(x) by Frequency(x) and set Frequency(x) = Frequency(x) + 1 aftervisiting solution x. The conclusion of Theorem A holds when CTS is based on frequency memory.

Additional enhancements that are possible using this TS methodology include the use of an associatedReverse Elimination Memory for streamlining the new tree search procedure associated with the TSprocess (which diers from the convergent TS approach that uses more exible memory). We alsopropose an approach for accelerating the classical tabu search Aspiration by Default rule in this setting,which may transform an exponential search into a much faster polynomial search. Finally, we givedesigns for more ecient forms of convergent tabu search in general.

3 Tabu Tree Search

In contrast to our previous use of the label T ime(x) for each solution x, however, we add the stipulationthat as soon as T ime(x) is assigned a value (i.e., as soon as x is visited), we do not permit its valueto be further changed. Accompanying this, we now reverse the Aspiration by Default rule, to requirethat, whenever all elements of N(x) have previously been visited, the method moves from x to the nodex N(x) that has the largest (rather than smallest) value of T ime(x), subject to the limitation thatthis value must be smaller than that of T ime(x) itself. The resulting method is as follows.

Tabu Tree Search (TTS)

1. From a given solution x, move to an unvisited neighbor x N(x) whenever possible, and stop ifthe label thus assigned to x is T ime(x) = |X |. Otherwise,

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MIC2001 - 4th Metaheuristics International Conference 335

2. Move to the visited neighbor x with the largest value of T ime(x) T ime(x).

We establish the relevant properties of the method as follows, under the assumption that the graph ofthe neighborhood space is connected.

Theorem BThe TTS method generates a tree, rooted at the initial solution, that spans the nodes of the neighbor-hood graph. Each edge of the tree is crossed exactly once in the direction away from the root, and atmost once in the direction toward the root. (No edges outside of the tree are crossed.) In addition:

(a) The unique path from any solution to the root is generated by repeatedly executing the rule ofStep 2 of the TTS method.

(b) Each time any solution x is visited, each labeled neighbor x of x is either an ancestor or descendantof x in the tree currently constructed (i.e., either x lies on the path from the root to x, or else xlies on the path from the root to x).

(c) Each time step 2 is executed to reach a visited node x, all nodes of the graph that are neighborsof visited nodes x, where T ime(x) > Time(x), are also visited nodes.

(d) Each time step 1 successfully identies an unvisited neighbor of x, then node x satises thecondition x = Argmax{T ime(y) : y is a node of the current tree and y has an unvisited neighbor}.

In common with the Aspiration by Default rule, the TTS approach in some cases may visit all solutionsby only visiting each solution a single time, hence eectively generating a Hamiltonian path throughthe neighborhood space, in contrast to the type of trajectory created by usual forms of tree search.However, more importantly, the TTS approach allows substantially greater exibility of choice thancustomary types of tree search, as embodied in branch and bound approaches. On the other hand, theTTS structure diers according to the choices made that is, dierent choices may produce dierentnumbers of revisited solutions (and, as previously remarked, some may produce no revisited solutions),thus producing trees of dierent topologies.

4 Conclusion

Theorems A and B can both readily be extended to apply to enhanced forms of the underlying TSdesigns. In summary, the key observations of our paper are: (1) strategic exibility is compatible withassured nite convergence, by special types of memory introduced in certain variants of tabu search;(2) the resulting search traverses the nodes of a graph in a signicantly dierent way than providedby tree search; (3) a simple tree search variant of the approach produces a type of tree search thatoers novel contrasts with branch and bound, and also diers notably from other tree searches such asreverse search and the Tarry Traverse.

References

[1] M. Andramonov. Some conditions of convergence of tabu search. Research report, Institute ofMathematics and Mechanics, Russia, 1999.

[2] F. Dammeyer and S. Voss. Dynamic tabu list management using the reverse elimination method.Ann. Oper. Res., 41:3146, 1993.

[3] H. Farreny. Completeness and admissibility for general heuristic search algorithms - a theoreticalstudy: Basic concepts and proofs. to appear in J. Heuristics.

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336 MIC2001 - 4th Metaheuristics International Conference

[4] F. Glover. Tabu search, part 2. ORSA J. Comput., 2:432, 1990.

[5] F. Glover and S. Hana. Composite tree searches for global optimization. Research report, Grad-uate School of Business and Administration, University of Colorado, Boulder, 1998.

[6] F. Glover and S. Hana. Tabu search and nite convergence. to appear in Special Issue of Discr.Appl. Math. on Foundations of heuristics in Combinatorial Optimization, 2000.

[7] F. Glover and M. Laguna. Tabu Search. Kluwer Academic Publishers, 1997.

[8] S. Hana. On the convergence of tabu search. J. Heuristics, 7:4758, 2000.

[9] S. Hana and A. Freville. Extension of reverse elimination method through a dynamic managementof the tabu list. to appear in R.A.I.R.O., 1999.

[10] G. Tarry. Le proble`me des labyrinthes. Nouvelles Annales de Mathematiques, 14(3):187190, 1895.

[11] G.L. Thompson. The tarry traverse, 1998. Class notes, Carnegie-Mellon University, GraduateSchool of Industrial Administration.

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