zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Variable neighborhood search: Principles and applications. (English) Zbl 0981.90063
Summary: Systematic change of neighborhood within a possibly randomized local search algorithm yields a simple and effective metaheuristic for combinatorial and global optimization, called Variable Neighborhood Search (VNS). We present a basic scheme for this purpose, which can easily be implemented using any local search algorithm as a subroutine. Its effectiveness is illustrated by solving several classical combinatorial or global optimization problems. Moreover, several extensions are proposed for solving large problem instances: using VNS within the successive approximation method yields a two-level VNS, called Variable Neighborhood Decomposition Search (VNDS); modifying the basic scheme to explore easily valleys far from the incumbent solution yields an efficient skewed VNS (SVNS) heuristic. Finally, we show how to stabilize column generation algorithms with help of VNS and discuss various ways to use VNS in graph theory, i.e., to suggest, disprove or give hints on how to prove conjectures, an area where metaheuristics do not appear to have been applied before.

90C59Approximation methods and heuristics
90C26Nonconvex programming, global optimization
90C27Combinatorial optimization
90B36Scheduling theory, stochastic
Full Text: DOI
[1] Anderberg, M. R.: Cluster analysis for application. (1973) · Zbl 0299.62029
[2] Beasley, J. E.: A note on solving large p-median problems. European journal of operational research 21, 270-273 (1985) · Zbl 0569.90021
[3] Bentley, J. L.: Fast algorithms for geometric traveling salesman problem. ORSA journal on computing 4, 387-411 (1992) · Zbl 0758.90071
[4] Boese, K. D.; Kahng, A. B.; Muddu, S.: A new adaptive multi-start technique for combinatorial global optimizations. Operational research letters 16, 101-113 (1994) · Zbl 0812.90126
[5] Brimberg, J.; Mladenović, N.: A variable neighborhood algorithm for solving the continuous location-allocation problem. Studies in location analysis 10, 1-12 (1996) · Zbl 0885.90069
[6] J. Brimberg, P. Hansen, N. Mladenović, É. Taillard, Improvements and comparison of heuristics for solving the multisource Weber problem, Operations Research 48 (3) (2000)
[7] Brinkmann, G.: Fast generation of cubic graphs. Journal of graph theory 23, No. 2, 139-149 (1996) · Zbl 0858.05093
[8] Caporossi, G.; Cvetković, D.; Gutman, I.; Hansen, P.: Variable neighborhood search for chemical graphs, part 2, graphs with extremal energy. Journal of chemical information and computer sciences 39, 984-996 (1999)
[9] Caporossi, G.; Hansen, P.: Variable neighborhood search for extremal graphs, 1. The autographix system. Discrete mathematics 212, 29-44 (2000) · Zbl 0947.90130
[10] Cooper, L.: Location--allocation problems. Operational research 11, 331-343 (1963) · Zbl 0113.14201
[11] Dorigo, M.; Maniezzo, V.; Colorni, A.: The ant system: optimization by a colony of cooperating agents. IEEE transactions on systems, man, and cybernetics -- part B 26, No. 1, 29-41 (1996)
[12] Du Merle, O.; Villeneuve, D.; Desrosiers, J.; Hansen, P.: Stabilized column generation. Discrete applied mathematics 194, 229-237 (1999) · Zbl 0949.90063
[13] Du Merle, O.; Hansen, P.; Jaumard, B.; Mladenović, N.: An interior point algorithm for minimum sum-of-squares clustering. SIAM journal on scientific computing 21, 1485-1505 (2000) · Zbl 1049.90129
[14] Feo, T.; Resende, M.: Greedy randomized adaptive search. Journal of global optimization 6, 109-133 (1995) · Zbl 0822.90110
[15] Fajtlowicz, S.: On conjectures of graffiti. Discrete mathematics 72, 113-118 (1987) · Zbl 0711.68081
[16] Fajtlowicz, S.: On conjectures of graffiti-II. Congressus numerantium 60, 187-197 (1987)
[17] Fajtlowicz, S.: On conjectures of graffiti-III. Congressus numerantium 66, 23-32 (1988)
[18] Fajtlowicz, S.: On conjectures of graffiti-IV. Congressus numerantium 70, 231-240 (1990)
[19] Fajtlowicz, S.: On conjectures of graffiti-V. Seventh international quadrennial conference on graph theory 1, 367-376 (1995) · Zbl 0843.05065
[20] R.A. Fisher, The use of multiple measurements in taxonomic problems, in Annual Eugenics VII, Part II, 1936, pp. 179--188
[21] Gendreau, M.; Hertz, A.; Laporte, G.: New insertion and postoptimization procedures for the traveling salesman problem. Operational research 40, 1086-1094 (1992) · Zbl 0767.90087
[22] Gendreau, M.; Hertz, A.; Laporte, G.: The traveling salesman problem with back-hauls. Computers and operational research 23, 501-508 (1996) · Zbl 0847.90135
[23] Glover, F.: Tabu search -- part II. ORSA jounal of computing 1, 190-206 (1989) · Zbl 0753.90054
[24] Glover, F.: Tabu search -- part II. ORSA journal of computing 2, 4-32 (1990) · Zbl 0771.90084
[25] F. Glover, M. Laguna, Tabu search, in: C. Reeves (Ed.), Modern Heuristic Techniques for Combinatorial Optimization, Blackwell, Oxford, 1993, pp. 70--150
[26] Glover, F.: Tabu thresholding: improved search by nonmonotonic trajectories. ORSA journal of computing 7, 426-442 (1995) · Zbl 0843.90097
[27] Glover, F.; Laguna, M.: Tabu search. (1997) · Zbl 0930.90083
[28] Gordon, A. D.: Classification: methods for the exploratory analysis of multivariate sata. (1981) · Zbl 0507.62057
[29] Gutman, I.; Polansky, O. E.: Mathematical concepts in organic chemistry. (1986) · Zbl 0657.92024
[30] Hansen, P.; Jaumard, B.: Algorithms for the maximum satisfiability problem. Computing 44, 279-303 (1990) · Zbl 0716.68077
[31] P. Hansen, B. Jaumard, S. Krau, O. du Merle, A stabilized column generation algorithm for the multisource Weber problem (in preparation)
[32] P. Hansen, B. Jaumard, N. Mladenović, A. Parreira, Variable neighborhood search for weighted maximum satisfiability (in preparation)
[33] Hansen, P.; Mladenović, N.: Variable neighborhood search for the p-median. Location science 5, No. 4, 207-226 (1998) · Zbl 0928.90043
[34] P. Hansen, N. Mladenović, J-MEANS, a new local search heuristic for minimum sum-of-squares clustering, Les Cahiers du GERAD G-99-14 and Pattern Recognition, forthcoming
[35] P. Hansen, N. Mladenović, An introduction to variable neighbourhood search. in: S. Voss et al. (Eds.), Metaheuristics, Advances and Trends in Local Search Paradigms for Optimization, Kluwer Academic Publishers, Dordrecht, 1999, pp. 433--458
[36] P. Hansen, N. Mladenović, D. Perez-Brito, Variable Neighborhood Decomposition Search, Journal of Heuristics, forthcoming
[37] Hertz, A.; Jaumard, B.; De Aragao, M. Poggi: Local optima topology for the k-coloring problem. Discrete applied mathematics 49, 257-280 (1994) · Zbl 0801.90116
[38] Holland, J. H.: Adaptation in natural and artificial systems. (1975) · Zbl 0317.68006
[39] Jancey, R. C.: Multidimensional group analysis. Australian journal of botany 14, 127-130 (1966)
[40] D.S. Johnson, L.A. McGeoch, The traveling salesman problem: a case study in local optimization, in: E.H.L. Aarts, J.K. Lenstra (Eds.), Local Search in Combinatorial Optimization, Wiley, London, 1997, pp. 215--310 · Zbl 0947.90612
[41] Kariv, O.; Hakimi, S. L.: An algorithmic approach to network location problems; part 2, the p-medians. SIAM journal on applied mathematics 37, 539-560 (1969) · Zbl 0432.90075
[42] Kirkpatrick, S.; Gelatt, C. D.; Vecchi, M.: Optimization by simulated annealing. Science 220, 671-680 (1983) · Zbl 1225.90162
[43] Kirkpatrick, S.; Toulouse, G.: Configuration space analysis of traveling salesman problems. Journal de physique 46, 1277-1292 (1985)
[44] S. Krau, Extensions du problème de Weber, Ph D. Thesis, École Polytechnique de Montréal (under direction of P. Hansen and B. Jaumard), 1997
[45] Kuenne, R. E.; Soland, R. M.: Exact and approximate solutions to the multisource Weber problem. Mathematical programming 3, 193-209 (1972) · Zbl 0245.90021
[46] Lin, S.: Computer solutions of the traveling salesman problem. Bell systems technical journal 44, 2245-2269 (1965) · Zbl 0136.14705
[47] Lin, S.; Kernighan, B. W.: An effective heuristic algorithm for the traveling salesman problem. Operational research 21, 498-516 (1973) · Zbl 0256.90038
[48] Macqueen, J. B.: Some methods for classification and analysis of multivariate observations. In: Proceedings of the fifth Berkeley symposium on mathematical statistics and probability 1, 281-297 (1967) · Zbl 0214.46201
[49] Mirchandani, P.; Francis, R.: Discrete location theory. (1990) · Zbl 0718.00021
[50] N. Mladenović, A variable neighborhood algorithm: A new metaheuristic for combinatorial optimization, Abstracts of papers presented at Optimization Days, Montréal, 1995, p. 112
[51] Mladenović, N.; Moreno, J. P.; Moreno-Vega, J.: A chain-interchange heuristic method. Yugoslav journal of operational research 6, No. 1, 41-54 (1996) · Zbl 0848.90104
[52] Mladenović, N.; Hansen, P.: Variable neighborhood search. Computers and operations research 24, 1097-1100 (1997) · Zbl 0889.90119
[53] Osman, I. H.; Christofides, N.: Capacitated clustering problems by hybrid simulated annealing and tabu search. International transactions on operational research 1, No. 3, 317-336 (1994) · Zbl 0857.90107
[54] Osman, I. H.; Laporte, G.: Metaheuristics: A bibliography. Annals of operational research 63, 513-628 (1996) · Zbl 0849.90097
[55] Papadimitriou, C. H.; Steiglitz, K.: Combinatorial optimization, algorithms and complexity. (1982) · Zbl 0503.90060
[56] Reeves, C.: Modern heuristic techniques for combinatorial problems. (1993) · Zbl 0942.90500
[57] Reinelt, G.: TSLIB -- a traveling salesman library. ORSA journal of computing 3, 376-384 (1991) · Zbl 0775.90293
[58] M.G.C. Resende, L.S. Pitsoulis, P.M. Pardalos, Approximate solution of weighted max-sat problems using GRASP, in: D. Du, J. Gu, P.M. Pardalos, (Eds.), Satisfiability Problem: Theory and Applications, DIMACS Series in Discrete Mathematics and Theoretical Computer Science 35, American Mathematical Society, Providence, RI, 1997 · Zbl 0889.68139
[59] Rolland, E.; Schilling, D. A.; Current, J. R.: An efficient tabu search procedure for the p-median problem. European journal of operational research 96, 329-342 (1996) · Zbl 0924.90102
[60] K.E. Rosing, Private communication
[61] Rosing, K. E.; Revelle, C. S.: Heuristic concentration: two stage solution construction. European journal of operational research 97, 75-86 (1997) · Zbl 0923.90107
[62] Rosing, K. E.; Revelle, C. S.; Rolland, E.; Schilling, D. A.; Current, J. R.: Heuristic concentration and tabu search: A head to head comparison. European journal of operational research 104, 93-99 (1998) · Zbl 0955.90056
[63] Späth, H.: Cluster dissection and analysis (Theory, Fortran programs, examples). (1985) · Zbl 0584.62094
[64] N. Thabet, Des algorithmes de génération de colonnes pour le problème de la p-mediane, Master thesis, École des HEC (under direction of P. Hansen), 1998
[65] Voss, S.: A reverse elimination approach for the p-median problem. Studies in locational analysis 8, 49-58 (1996) · Zbl 1176.90365
[66] Whitaker, R.: A fast algorithm for the greedy interchange for large-scale clustering and median location problems. Infor 21, 95-108 (1863) · Zbl 0527.90017