Dynamical System Approaches to Combinatorial Optimization

  • Jens Starke
Reference work entry


Several dynamical system approaches to combinatorial optimization problems are described and compared. These include dynamical systems derived from penalty methods; the approach of Hopfield and Tank; self-organizing maps, that is, Kohonen networks; coupled selection equations; and hybrid methods. Many of them are investigated analytically, and the costs of the solutions are compared numerically with those of solutions obtained by simulated annealing and the costs of a global optimal solution.

Using dynamical systems, a solution to the combinatorial optimization problem emerges in the limit of large times as an asymptotically stable point of the dynamics. The obtained solutions are often not globally optimal but good approximations of it. Dynamical system and neural network approaches are appropriate methods for distributed and parallel processing. Because of the parallelization, these techniques are able to compute a given task much faster than algorithms which are using a traditional sequentially working digital computer.

This chapter focuses on dynamical system approaches to the linear two-index assignment problem and the \(\mathcal{N}\mathcal{P}\)-hard three-index assignment problem. These and extensions thereof can be used as models for many industrial problems like manufacturing planning and optimization of flexible manufacturing systems. This is illustrated for an example in distributed robotic systems.


Simulated Annealing Assignment Problem Travel Salesman Problem Travel Salesman Problem Combinatorial Optimization Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Recommended Reading

  1. 1.
    Y. Abu-Mostafa, D. Psaltis, Optical neural computers. Sci. Am. 256(3), 66–73 (1987)Google Scholar
  2. 2.
    H. Achatz, P. Kleinschmidt, K. Paparrizos, A dual forest algorithm for the assignment problem, in The Victor Klee Festschrift. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 4 (American Mathematical Society, Providence, 1991), pp. 1–10Google Scholar
  3. 3.
    S. Amari, Mathematical foundations of neurocomputing, in Proceedings of the IEEE, vol. 78 (IEEE, New York, 1990), pp. 1443–1463Google Scholar
  4. 4.
    J. Anderson, E. Rosenfeld, Neurocomputing, Foundations of Research (MIT, Cambridge, 1988)Google Scholar
  5. 5.
    J. Anderson, A. Pellionisz, E. Rosenfeld, Neurocomputing 2, Directions for Research (MIT, Cambridge, 1990)Google Scholar
  6. 6.
    B. Angèniol, G. De la Croix Vaubois, J.-Y. Le Texier, Self-organizing feature maps and the travelling salesman problem. Neural Netw. 1, 289–293 (1988)Google Scholar
  7. 7.
    D. Anosov, I. Bronshtein, S. Aranson, V. Grines, Smooth dynamical systems, in Dynamical Systems I. Encyclopaedia of Mathematical Sciences, vol. 1 (Springer, Heidelberg/Berlin/ New York, 1988), pp. 149–233Google Scholar
  8. 8.
    R. Arkin, Behavior-Based Robotics (MIT, Cambridge/London, 1998)Google Scholar
  9. 9.
    V.I. Arnol’d, Gewöhnliche Differentialgleichungen (Deutscher Verlag der Wissenschaften, Berlin, 1979/1991)Google Scholar
  10. 10.
    V.I. Arnol’d, Geometrische Methoden in der Theorie der gewöhnlichen Differentialgleichungen (Deutscher Verlag der Wissenschaften, Berlin, 1987)Google Scholar
  11. 11.
    V.I. Arnol’d, Yu. S. Il’yashenko, Ordinary differential equations, in Dynamical Systems I, ed. by D. Anosov, V. Arnol’d. Encyclopaedia of Mathematical Sciences, vol. 1 (Springer, Berlin/Heidelberg/New York, 1988), pp. 1–148Google Scholar
  12. 12.
    H. Asama, T. Arai, T. Fukuda, T. Hasegawa (eds.), Distributed Autonomous Robotic System (DARS 5) (Springer, Heidelberg/Berlin/New York, 2002)Google Scholar
  13. 13.
    M. Avriel, Nonlinear Programming – Analysis and Methods (Prentice-Hall, Englewood Cliffs, 1976)MATHGoogle Scholar
  14. 14.
    B. Baird, Bifurcation and category learning in network models of oscillating cortex. Physica D 42, 365–384 (1990)Google Scholar
  15. 15.
    W. Banzhaf, A new dynamical approach to the travelling salesman problem. Phys. Lett. A 136(1, 2), 45–51 (1989)Google Scholar
  16. 16.
    W. Banzhaf, The molecular traveling salesman. Biol. Cybern. 64, 7–14 (1990)MATHGoogle Scholar
  17. 17.
    M. Becht, T. Buchheim, P. Burger, G. Hetzel, G. Kindermann, R. Lafrenz, N. Oswald, M. Schanz, M. Schulé, P. Molnar, J. Starke, P. Levi, Three-index assignment of robots to targets: an experimental verification, in Proceedings of the 6th International Conference on Intelligent Autonomous Systems (IAS-6), ed. by E. Pagello et al. (IOS, Amsterdam/Washington, DC, 2000), pp. 156–163Google Scholar
  18. 18.
    M. Bestehorn, H. Haken, Associative memory of a dynamical system: the example of the convection instability. Z. Phys. B 82, 305–308 (1991)Google Scholar
  19. 19.
    K. Binder, A. Young, Spin glasses: experimental facts, theoretical concepts, and open questions. Rev. Mod. Phys. 58(4), 801–963 (1986)Google Scholar
  20. 20.
    A.M. Bloch, A. Iserles, On the optimality of double-bracket flows. Int. J. Math. Math. Sci. 2004(61–64), 3301–3319 (2004)MathSciNetMATHGoogle Scholar
  21. 21.
    I. Bomze, Evolution towards the maximum clique. J. Glob. Optim. 10, 143–164 (1997)MathSciNetMATHGoogle Scholar
  22. 22.
    I. Bomze, M. Budinich, P. Pardalos, M. Pelillo, The maximum clique problem, in Handbook of Combinatorial Optimization, ed. by D.-Z. Du, P.M. Pardalos (Kluwer, Dordrecht/Boston/London, 1999)Google Scholar
  23. 23.
    I. Bomze, M. Pelillo, V. Stix, Approximating the maximum weight clique using replicator dynamics. IEEE Trans. Neural Netw. 11(6), 1228–1241 (2000)Google Scholar
  24. 24.
    I. Bomze, M. Budinich, M. Pelillo, C. Rossi, Annealed replication: a new heuristic for the maximum clique problem. Discret. Appl. Math. 121, 27–49 (2002)MathSciNetMATHGoogle Scholar
  25. 25.
    R. Brockett, Dynamical systems that sort lists, diagonalize matrices and solve linear programming problems, in Proceedings of the 27th Conference on Decision and Control (IEEE, New York, 1988), pp. 799–803Google Scholar
  26. 26.
    R. Brockett, W. Wong, A gradient flow for the assignment problem, in New Trends in Systems Theory, ed. by G. Conte, A. Perdon, B. Wyman (Birkhäuser, Boston/Basel/Berlin, 1991), pp. 170–177Google Scholar
  27. 27.
    R. Brooks, New approaches to robotics. Science 253, 1227–1232 (1991)Google Scholar
  28. 28.
    R. Burkard, Methoden der Ganzzahligen Optimierung (Springer, Wien/New York, 1972)MATHGoogle Scholar
  29. 29.
    R. Burkard, M. Dell’Amico, S. Martello, Assignment Problems (Society for Industrial and Applied Mathematics, Philadelphia, 2009)MATHGoogle Scholar
  30. 30.
    D. Cvijović, J. Klinowski, Taboo search: an approach to the multiple minima problem. Science 267(3), 664–666 (1995)MathSciNetGoogle Scholar
  31. 31.
    A. Daffertshofer, How do ensembles occupy space? Eur. Phys. J. Spec. Top. 157, 79–91 (2008)Google Scholar
  32. 32.
    A. Daffertshofer, H. Haken, J. Portugali, Self-organized settlements. Environ. Plan. B Plan. Des. 28(1), 89–102 (2001)Google Scholar
  33. 33.
    L. Davis, Handbook of Genetic Algorithms (Van Nostrand Reinhold, New York, 1991)Google Scholar
  34. 34.
    G. Di Marzo Serugendo, A. Karageorgos, O.F. Rana, F. Zambonelli (eds.), Engineering Self-Organising Systems – Nature-Inspired Approaches to Software Engineering (Springer, Heidelberg/Berlin/New York, 2004)MATHGoogle Scholar
  35. 35.
    R. Durbin, D. Willshaw, An analogue approach to the travelling salesman problem using an elastic net method. Nature 326, 689–691 (1987)Google Scholar
  36. 36.
    W. Ebeling, Self-organization, valuation and optimization, in On Self-Organization, ed. by R. Mishra, D. Maaß, E. Zwierlein. Springer Series in Synergetics, vol. 61 (Springer, Berlin/Heidelberg, 1994), pp. 185–196Google Scholar
  37. 37.
    W. Ebeling, A. Engel, R. Feistel, Physik der Evolutionsprozesse (Akademie, Berlin, 1990)MATHGoogle Scholar
  38. 38.
    M. Eigen, P. Schuster, The hypercycle – Part A: emergence of the hypercycle. Die Naturwissenschaften 64, 541–565 (1977)Google Scholar
  39. 39.
    M. Eigen, P. Schuster, The hypercycle – Part B: the abstract hypercycle. Die Naturwissenschaften 65, 7–41 (1978)Google Scholar
  40. 40.
    H. Eiselt, G. Pederzoli, C.-L. Sandblom, Continuous Optimization Models – Operations Research (Walter de Gruyter, Berlin/New York, 1987)Google Scholar
  41. 41.
    F. Tay, Contingency management in flexible manufacturing systems using modal state logic. J. Manuf. Syst. 18(5), 345–357 (1999)MathSciNetGoogle Scholar
  42. 42.
    J. Fort, Solving a combinatorial problem via self-organizing process: an application of the kohonen algorithm to the traveling salesman problem. Biol. Cybern. 59, 33–40 (1988)MathSciNetMATHGoogle Scholar
  43. 43.
    T.D. Frank, On a multistable competitive network model in the case of an inhomogeneous growth rate spectrum: with an application to priming. Phys. Lett. A 373(45), 4127–4133 (2009)MATHGoogle Scholar
  44. 44.
    T.D. Frank, Multistable selection equations of pattern formation type in the case of inhomogeneous growth rates: with applications to two-dimensional assignment problems. Phys. Lett. A 375(12), 1465–1469 (2011)MATHGoogle Scholar
  45. 45.
    T. Fukuda, S. Nakagawa, Approach to the dynamically reconfigurable robotic system. J. Intell. Robot. Syst. 1(1), 55–72 (1988)Google Scholar
  46. 46.
    T. Fukuda, T. Ueyama, Cellular Robotics and Micro Robotic Systems. World Scientific Series in Robotics and Automated Systems, vol. 10 (World Scientific, Singapore/New Jersey/ Hong Kong, 1994)Google Scholar
  47. 47.
    M. Garey, D. Johnson, Computers and Intractability (Freeman and Company, San Francisco, 1979)MATHGoogle Scholar
  48. 48.
    A. Gee, S. Aiyer, R. Prager, An analytical framework for optimizing neural networks. Neural Netw. 6, 79–97 (1993)Google Scholar
  49. 49.
    F. Glover, Tabu search – Part I. ORSA J. Comput. 1, 190–206 (1989)MathSciNetMATHGoogle Scholar
  50. 50.
    F. Glover, Tabu search – Part II. ORSA J. Comput. 2, 4–32 (1989)MathSciNetGoogle Scholar
  51. 51.
    F. Glover, E. Taillard, D. de Werra, Tabu Search. Annals of Operations Research, vol. 41 (J.C. Baltzer, Basel, 1993), pp. 3–28Google Scholar
  52. 52.
    D. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning (Addison-Wesley, Reading, 1989)MATHGoogle Scholar
  53. 53.
    R. Graham, M. Grötschel, L. Lovász, Handbook of Combinatorics (Elsevier Science B. V., Amsterdam/Lausanne/New York, 1995)MATHGoogle Scholar
  54. 54.
    C. Großmann, J. Terno, Numerik der Optimierung. Teubner Studienbücher: Mathematik (Teubner, Stuttgart, 1993)MATHGoogle Scholar
  55. 55.
    M. Grötschel, L. Lovász, Combinatorial optimization, in Handbook of Combinatorics [53], chapter 28, pp. 1541–1597Google Scholar
  56. 56.
    C. Guus, E. Boender, H. Edwin Romeijn, Stochastic methods, in Handbook of Global Optimization, ed. by R. Horst, P. Pardalos (Kluwer, Dordrecht/Boston/London, 1995), pp. 829–869Google Scholar
  57. 57.
    H. Haken, Pattern formation and pattern recognition – an attempt at a synthesis, in Pattern Formation by Dynamic Systems and Pattern Recognition, ed. by H. Haken. Springer Series in Synergetics, vol. 5 (Springer, Heidelberg/Berlin/New York, 1979), pp. 2–13Google Scholar
  58. 58.
    H. Haken, Advanced Synergetics. Springer Series in Synergetics (Springer, Heidelberg/Berlin/New York, 1983)Google Scholar
  59. 59.
    H. Haken, Synergetics, An Introduction. Springer Series in Synergetics (Springer, Heidelberg/Berlin/New York, 1983)Google Scholar
  60. 60.
    H. Haken, Synergetic Computers and Cognition – A Top-Down Approach to Neural Nets. Springer Series in Synergetics (Springer, Heidelberg/Berlin/New York, 1991)Google Scholar
  61. 61.
    H. Haken, Principles of Brain Functioning – A Synergetic Approach to Brain Activity, Behavior and Cognition. Springer Series in Synergetics (Springer, Berlin/Heidelberg/ New York, 1996)Google Scholar
  62. 62.
    H. Haken, Decision making and optimization in regional planning, in Knowledge and Networks in a Dynamic Economy, ed. by M. Beckmann, B. Johansson, F. Snickars, R. Thord (Springer, Berlin/Heidelberg/New York, 1998)Google Scholar
  63. 63.
    H. Haken, M. Schanz, J. Starke, Treatment of combinatorial optimization problems using selection equations with cost terms – Part I: two-dimensional assignment problems. Physica D 134, 227–241 (1999)MathSciNetMATHGoogle Scholar
  64. 64.
    U. Helmke, J.B. Moore, Optimization and Dynamical Systems (Springer, London/ Berlin/Heidelberg, 1994)Google Scholar
  65. 65.
    J. Hertz, A. Krogh, R. Palmer, Introduction to the Theory of Neural Computation (Addison-Wesley Publishing Company, Redwood City, 1991)Google Scholar
  66. 66.
    M. Hestenes, Optimization Theory (Wiley, New York/London, 1975)MATHGoogle Scholar
  67. 67.
    M. Hirsch, B. Baird, Computing with dynamic attractors in neural networks. BioSystems 34, 173–195 (1995)Google Scholar
  68. 68.
    M. Hirsch, S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra (Academic, New York, 1974)MATHGoogle Scholar
  69. 69.
    J. Hofbauer, K. Sigmund, The Theory of Evolution and Dynamical Systems. London Mathematical Society Student Texts, vol. 7 (Cambridge University Press, Cambridge/ New York, 1988)MATHGoogle Scholar
  70. 70.
    J. Holland, Adaption in Natural and Artificial Systems (University of Michigan Press, Ann Arbor, 1975)MATHGoogle Scholar
  71. 71.
    J. Hopfield, Neural networks and physical systems with emergent collective computational abilities. Proc. Natl. Acad. Sci. 79, 2554–2558 (1982)MathSciNetGoogle Scholar
  72. 72.
    J. Hopfield, Neurons with graded response have collective computational properties like those of two-state neurons. Proc. Natl. Acad. Sci. 81, 3088–3092 (1984)Google Scholar
  73. 73.
    J. Hopfield, D. Tank, Neural computation of decisions in optimization problems. Biol. Cybern. 52, 141–152 (1985)MathSciNetMATHGoogle Scholar
  74. 74.
    J. Hopfield, D. Tank, Computing with neural circuits: a model. Science 233, 625–633 (1986)Google Scholar
  75. 75.
    R. Horst, Nichtlineare Optimierung (Carl Hanser, München/Wien, 1979)MATHGoogle Scholar
  76. 76.
    T. Kaga, J. Starke, P. Molnár, M. Schanz, T. Fukuda, Dynamic robot-target assignment – dependence of recovering from breakdowns on the speed of the selection process, in Distributed Autonomous Robotic Systems (DARS 4), ed. by L.E. Parker, G. Bekey, J. Barhen (Springer, Heidelberg/New York/Tokyo, 2000), pp. 325–334Google Scholar
  77. 77.
    B. Kamgar-Parsi, B. Kamgar-Parsi, On problem solving with Hopfield neural networks. Biol. Cybern. 62, 415–423 (1990)MathSciNetMATHGoogle Scholar
  78. 78.
    N. Karmarkar, A new polynomial-time algorithm for linear programming. Combinatorica 4, 373–395 (1984)MathSciNetMATHGoogle Scholar
  79. 79.
    L.E. Kavraki, J.-C. Latombe, Probabilistic roadmaps for robot path planning, in Practical Motion Planning in Robotics: Current Approaches and Future Directions, ed. by K. Gupta, A.P. del Pobil (Wiley, Chichester/New York, 1998), pp. 33–53Google Scholar
  80. 80.
    W. Kinzel, Spin glasses and memory. Phys. Scr. 35, 398–401 (1987)MathSciNetMATHGoogle Scholar
  81. 81.
    S. Kirkpatrick, Optimization by simulated annealing: quantitative studies. J. Stat. Phys. 34(5/6), 975–986 (1984)MathSciNetGoogle Scholar
  82. 82.
    S. Kirkpatrick, G. Toulouse, Configuration space analysis of travelling salesman problems. J. Phys. 46, 1277–1292 (1985)MathSciNetGoogle Scholar
  83. 83.
    S. Kirkpatrick, C. Gelatt, M. Vecchi, Optimization by simulated annealing. Science 220(4598), 671–680 (1983)MathSciNetMATHGoogle Scholar
  84. 84.
    T. Kohonen, Self-Organization and Associative Memory (Springer, Berlin/Heidelberg/ New York, 1984)MATHGoogle Scholar
  85. 85.
    T. Kohonen, Self-Organizing Maps (Springer, Berlin/Heidelberg/New York, 1995)Google Scholar
  86. 86.
    M.J.B. Krieger, J.-B. Billeter, L. Keller, Ant-like task allocation and recruitment in cooperative robots. Nature 406, 992–995 (2000)Google Scholar
  87. 87.
    A. Kusiak, Flexible manufacturing systems: a structural approach. Int. J. Prod. Res. 23(6), 1057–1073 (1985)Google Scholar
  88. 88.
    J.-C. Latombe, Robot Motion Planning, 3rd edn. (Kluwer, Dordrecht/Boston/London, 1993)Google Scholar
  89. 89.
    D. Luenberger, Introduction to Linear and Nonlinear Programming (Addison-Wesley Publishing Company, New York/London, 1973)MATHGoogle Scholar
  90. 90.
    S. Matsuda, Stability of solutions in Hopfield neural network. Syst. Comput. Jpn 26(5), 67–78 (1995) (Translated from Vol. J77-D-II, No. 7, July 1994, pp. 1366–1374)Google Scholar
  91. 91.
    S. Matsuda, Theoretical considerations on the capabilities of crossbar switching by Hopfield networks, in Proceedings of the 1995 IEEE International Conference on Neural Networks (IEEE, 1995), pp. 1107–1110Google Scholar
  92. 92.
    N. Metropolis, M. Rosenbluth, A. Teller, E. Teller, Equation of state calculations by fast computing machines. J. Chem. Phys. 21(6), 1087–1092 (1953)Google Scholar
  93. 93.
    Z. Michalewicz, Genetic Algorithms + Data Structures = Evolution Programs (Springer, Berlin/Heidelberg/New York, 1992)MATHGoogle Scholar
  94. 94.
    P. Molnár, J. Starke, Control of distributed autonomous robotic systems using principles of pattern formation in nature and pedestrian behaviour. IEEE Trans. Syst. Men Cybern. Part B 31(3), 433–436 (2001)Google Scholar
  95. 95.
    B. Müller, J. Reinhardt, Neural Networks – An Introduction (Springer, Berlin/Heidelberg/ New York, 1991)Google Scholar
  96. 96.
    Y. Nesterov, Interior-point methods: an old and new approach to nonlinear programming. Math. Program. 79, 285–297 (1997)MathSciNetMATHGoogle Scholar
  97. 97.
    R. Neubecker, G.-L. Oppo, B. Thuering, T. Tschudi, Pattern formation in a liquid-crystal light valve with feedback, including polarization, saturation, and internal threshold effects. Phys. Rev. A 52(1), 791–808 (1995)Google Scholar
  98. 98.
    G. Nicolis, I. Prigogine, Self-Organization in Non-Equilibrium Systems (Wiley, New York, 1977)Google Scholar
  99. 99.
    K. Pál, Genetic algorithms for the traveling salesman problem based on a heuristic crossover operation. Biol. Cybern. 69, 539–546 (1993)MATHGoogle Scholar
  100. 100.
    C. Papadimitriou, K. Steiglitz, Combinatorial Optimization – Algorithms and Complexity (Prentice-Hall, Englewood Cliffs, 1982)MATHGoogle Scholar
  101. 101.
    M. Pelillo, Replicator equations, maximal cliques, and graph isomorphism. Neural Comput. 11, 1933–1955 (1999)Google Scholar
  102. 102.
    M. Pelillo, Evolutionary game dynamics in combinatorial optimization: an overview, in Proceedings of the EvoWorkshops on Applications of Evolutionary Computing, ed. by E.J.W. Boers, J. Gottlieb, P.L. Lanzi, R.E. Smith, S. Cagnoni, E. Hart, G.R. Raidl, H. Tijink (Springer, Heidelberg/New York/Tokyo, 2001), pp. 182–192Google Scholar
  103. 103.
    M. Pelillo, Evolutionary game dynamics in combinatorial optimization: an overview, in Applications of Evolutionary Computing, ed. by E. Boers. Lecture Notes in Computer Science, vol. 2037 (Springer, Berlin/Heidelberg, 2001), pp. 182–192Google Scholar
  104. 104.
    M. Pelillo, K. Siddiqi, S.W. Zucker, Matching hierarchical structures using association graphs. IEEE Trans. Pattern Anal. Mach. Intell. 21(11), 1105–1120 (1999)Google Scholar
  105. 105.
    P. Peretto, Neural networks and combinatorial optimization, in Proceedings of the International Conference “Les Entretiens de Lyon” (Springer, Paris, 1990), pp. 127–134Google Scholar
  106. 106.
    C. Peterson, B. Söderberg, Neural optimization, in Brain Theory and Neural Networks, ed. by M. Arbib (MIT, Cambridge/London, 1995), pp. 617–621Google Scholar
  107. 107.
    W. Press, S. Teukolsky, W. Vetterling, B. Flannery, Numerical Recipes in C (Cambridge University Press, Cambridge/New York/1992)MATHGoogle Scholar
  108. 108.
    D. Psaltis, D. Brady, X. Gu, S. Lin, Holography and artificial neural networks. Nature 343, 325–330 (1990)Google Scholar
  109. 109.
    I. Rechenberg, Evolutionsstrategie (Friedrich Frommann, Stuttgart Bad Cannstatt, 1973)Google Scholar
  110. 110.
    C. Robinson, Dynamical Systems – Stability, Symbolic Dynamics, and Chaos (CRC, Boca Raton/Ann Arbor/London, 1995)MATHGoogle Scholar
  111. 111.
    H.-P. Schwefel, Numerische Optimierung von Computer-Modellen mittels der Evolutionsstrategie (Birkhäuser, Basel/Stuttgart, 1977)MATHGoogle Scholar
  112. 112.
    Z. Simeu-Abazi, C. Sassine, Maintenance integration in manufacturing systems: from the modeling tool to evaluation. Int. J. Flex. Manuf. Syst. 13(3), 267–285 (2001)Google Scholar
  113. 113.
    K. Smith, Neural networks for combinatorial optimization: a review of more than a decade of research. INFORMS J. Comput. 11(1), 15–34 (1999)MathSciNetMATHGoogle Scholar
  114. 114.
    P. Spellucci, Numerische Verfahren der nichtlinearen Optimierung (Birkhäuser, Basel/Boston/Berlin, 1993)MATHGoogle Scholar
  115. 115.
    F.C.R. Spieksma, Multi-index assignment problems: complexity, approximation, applications, in Nonlinear Assignment Problems: Algorithms and Applications, ed. by L. Pitsoulis, P. Pardalos (Kluwer, Amsterdam, 2000), pp. 1–12Google Scholar
  116. 116.
    J. Starke, Cost oriented competing processes – a new handling of assignment problems, in System Modelling and Optimization, ed. by J. Doležal, J. Fidler (Chapman & Hall, London/Glasgow, 1996), pp. 551–558Google Scholar
  117. 117.
    J. Starke, Combinatorial optimization based on the principles of competing processes, in Self-Organization of Complex Structures: From Individual to Collective Dynamics. Part I: Evolution of Complexity and Evolutionary Optimization, ed. by F. Schweitzer (Gordon and Breach, London, 1997), pp. 165–178Google Scholar
  118. 118.
    J. Starke, Kombinatorische Optimierung auf der Basis gekoppelter Selektionsgleichungen. Ph.D. thesis, Universität Stuttgart, Verlag Shaker, Aachen, 1997Google Scholar
  119. 119.
    J. Starke, Dynamical assignments of distributed autonomous robotic systems to manufacturing targets considering environmental feedbacks, in Proceedings of the 17th IEEE International Symposium on Intelligent Control (ISIC’02), Vancouver, 2002, pp. 678–683Google Scholar
  120. 120.
    J. Starke, C. Ellsässer, T. Fukuda, Self-organized control in cooperative robots using a pattern formation principle. Phys. Lett. A 375, 2094–2098 (2011)MATHGoogle Scholar
  121. 121.
    J. Starke, P. Molnár, Dynamic control of distributed autonomous robotic systems with underlying three-index assignments, in Proceedings of the IECON 2000 (IEEE, New York, 2000), pp. 2093–2098Google Scholar
  122. 122.
    J. Starke, M. Schanz, Dynamical system approaches to combinatorial optimization, in Handbook of Combinatorial Optimization, vol. 2, ed. by D.-Z. Du, P. Pardalos (Kluwer, Dordrecht/Boston/London, 1998), pp. 471–524Google Scholar
  123. 123.
    J. Starke, M. Schanz, H. Haken, Self-organized behaviour of distributed autonomous mobile robotic systems by pattern formation principles, in Distributed Autonomous Robotic Systems (DARS 3), ed. by T. Lueth, R. Dillmann, P. Dario, H. Wörn (Springer, Heidelberg/Berlin/ New York, 1998), pp. 89–100Google Scholar
  124. 124.
    J. Starke, M. Schanz, H. Haken, Treatment of combinatorial optimization problems using selection equations with cost terms – Part II: three-dimensional assignment problems. Physica D 134, 242–252 (1999)MathSciNetMATHGoogle Scholar
  125. 125.
    J. Starke, T. Kaga, M. Schanz, T. Fukuda, Experimental study on self-organized and error resistant control of distributed autonomous robotic systems. Int. J. Robot. Res. 24, 465–486 (2005)Google Scholar
  126. 126.
    G.A. Tagliarini, J.F. Christ, E.W. Page, Optimization using neural networks. IEEE Trans. Comput. 40(12), 1347–1358 (1991)Google Scholar
  127. 127.
    T.-Y. Tam, Gradient flows and double bracket equations. Differ. Geom. Appl. 20, 209–224 (2004)MATHGoogle Scholar
  128. 128.
    D. Tank, J. Hopfield, Simple neural optimization networks: an A/D converter, signal decision circuit and a linear programming circuit. IEEE Trans. Circuits Syst. CAS-33(5), 533–541 (1986)Google Scholar
  129. 129.
    K. Tsuchiya, T. Nishiyama, K. Tsujita, A deterministic annealing algorithm for a combinatorial optimization problem by the use of replicator equations, in IEEE International Conference on Systems, Man, and Cybernetics, 1999. Conference Proceedings, Tokyo, vol. 1, 1999, pp. 256–261Google Scholar
  130. 130.
    K. Tsuchiya, T. Nishiyama, K. Tsujita, A deterministic annealing algorithm for a combinatorial optimization problem using replicator equations. Physica D 149, 161–173 (2001)MathSciNetMATHGoogle Scholar
  131. 131.
    Y. Uesaka, Mathematical aspects of neuro-dynamics for combinatorial optimization. IEICE Trans. E 74(6), 1368–1372 (1991)Google Scholar
  132. 132.
    K. Urahama, Analog circuit for solving assignment problems. IEEE Trans. Circuits Syst. 41(5), 426–429 (1994)MATHGoogle Scholar
  133. 133.
    D. Van den Bout, T. Miller, A traveling salesman objective function that works, in Proceedings of the IEEE International Conference on Neural Networks 1988, vol. II (IEEE, San Diego, 1988), pp. II–299–II–303Google Scholar
  134. 134.
    D. Van den Bout, T. Miller III, Improving the performance of the Hopfield-Tank neural network through normalization and annealing. Biol. Cybern. 62, 129–139 (1989)Google Scholar
  135. 135.
    P. van Laarhoven, E. Aarts, Simulated Annealing: Theory and Applications (Reidel Publishing Company, Dordrecht/Boston/Lancaster/Tokyo, 1987)MATHGoogle Scholar
  136. 136.
    S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos (Springer, Berlin/Heidelberg/New York, 1990)MATHGoogle Scholar
  137. 137.
    G. Wilson, G. Pawley, On the stability of the travelling salesman problem algorithm of Hopfield and Tank. Biol. Cybern. 58, 63–70 (1988)MathSciNetMATHGoogle Scholar
  138. 138.
    W. Wong, Matrix representation and gradient flows for NP-hard problems. J. Optim. Theory Appl. 87(1), 197–220 (1995)MathSciNetMATHGoogle Scholar
  139. 139.
    A. Yuille, Constrained optimization and the elastic net, in Brain Theory and Neural Networks, ed. by M. Arbib (MIT, Cambridge/London, 1995), pp. 250–255Google Scholar
  140. 140.
    M.M. Zavlanos, G.J. Pappas, A dynamical system approach to weighted graph matching. Automatica 44, 2817–2824 (2008)MathSciNetMATHGoogle Scholar

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© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Mathematics & Computer ScienceTechnical University of DenmarkKongens LyngbyDenmark

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