Nonlinear Programming in Telecommunications

  • Athanasios Migdalas


Telecommunications have always been the subject of application for advanced mathematical techniques. In this chapter, we review classical nonlinear programming approaches to modeling and solving certain problems in telecommunications. We emphasize the common aspects of telecommunications and road networks, and indicate that several lessons are to be learned from the field of transportation science, where game theoretic and equilibrium approaches have been studied for more than forty years. Several research directions are also stated.


Nonlinear optimization Frank-Wolfe-like algorithms simplicial decomposition team games routing equilibrium flows capacity assignment network design bilevel programming Nash equilibrium Wardrop’s principle Stackelberg game 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. E. Altman, R. El Azouzi, and V. Abramov. Non-cooperative routing in loss networks. Performance Evaluation, 49:43–55, 2002.MATHCrossRefGoogle Scholar
  2. E. Altman, T. Başar, T. Jiménez, and N. Shikin. Competitive routing in networks with polynomial costs. Technical report, INRIA, B.P. 93, 06902 Sophia Antipolis Cedex, France, 2000.Google Scholar
  3. M.S. Bazaraa and C.M. Shetty. Nonlinear programming — Theory and algorithms. John Wiley and Sons, New York, 1979.MATHGoogle Scholar
  4. N.G. Beans, F.P. Kelly, and P.G. Taylor. Braess’s paradox in a loss network. Journal of Applied Prob., 34:155–159, 1997.CrossRefGoogle Scholar
  5. M. Beckmann, C.B. McGuire, and C.B. Winsten. Studies in Economics of Transportation. Yale University Press, 1956.Google Scholar
  6. K.P. Bennet. Global tree optimization: A non-greedy decision tree algorithm. Computing Science and Statistics, 26:156–160, 1994.Google Scholar
  7. K.P. Bennet and O.L. Mangasarian. Robust linear programming discrimination of two linearly inseparable sets. Optimization Methods and Software, 1:23–34, 1992.CrossRefGoogle Scholar
  8. K.P. Bennet and O.L. Mangasarian. Bilinear separation of two sets in n-space. Computational Optimization and Applications, 2:207–227, 1993.CrossRefMathSciNetGoogle Scholar
  9. J. Berechman. Highway-capacity utilization and investment in transportation corridors. Environment and Planning, 16A: 1475–1488, 1984.CrossRefGoogle Scholar
  10. F. Berggren. Power control and adaptive resource allocation in DS-CDMA systems. PhD thesis, Royal Institute of Technology, 2003.Google Scholar
  11. D. Bernstein and S.A. Gabriel. Solving the nonadditive traffic equilibrium problem. Technical Report SOR-96-14, Statistics and Operations Research Series, Princeton University, 1996.Google Scholar
  12. D.P. Bertsekas. On the Goldstein-Levitin-Polyak gradient projection method. IEEE Trans. Automat. Control, AC-21:174–184, 1976.CrossRefMathSciNetGoogle Scholar
  13. D.P. Bertsekas. Constrained optimization and Lagrange multiplier methods. Academic Press, New York, 1982a.MATHGoogle Scholar
  14. D.P. Bertsekas. Projected Newton methods for optimization problems with simple constraints. SIAM Journal on Control and Optimization, 20:221–246, 1982b.MATHCrossRefMathSciNetGoogle Scholar
  15. D.P. Bertsekas and E.M. Gafni. Projected Newton methods and optimization of multicommodity flows. IEEE Trans. Automat. Control, AC-28:1090–1096, 1983.CrossRefMathSciNetGoogle Scholar
  16. D.P. Bertsekas, E.M. Gafni, and R.G. Gallager. Second derivative algorithms for minimum delay distributed routing in networks. IEEE Trans. Comm., COM-32:911–919, 1984.CrossRefMathSciNetGoogle Scholar
  17. D.P. Bertsekas and R. Gallager. Data networks. Prentice-Hall, Englewood Cliffs, NJ, 1992.MATHGoogle Scholar
  18. D.P. Bertsekas and J.N. Tsitsiklis. Parallel and distributed computation — Numerical methods. Prentice-Hall, 1989.Google Scholar
  19. D. Bienstock and O. Raskina. Aymptotic analysis of the flow deviation method for the maximum concurrent flow problem. Mathematical Programming, 91:479–492, 2002.MATHCrossRefMathSciNetGoogle Scholar
  20. D. Bienstock and I. Saniee. ATM network design: Traffic models and optimization-based heuristics. Technical Report 98-20, DIMACS, 1998.Google Scholar
  21. J.A. Blue and K.P Bennett. Hybrid extreme point tabu search. Technical Report 240, Dept of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY, 1996.Google Scholar
  22. N.L. Boland, A.T Ernst, C.J. Goh, and A.I. Mees. A faster version of the ASG algorithm. Applied Mathematics Letters, 7:23–27, 1994.MATHCrossRefMathSciNetGoogle Scholar
  23. N.L. Boland, C.J. Goh, and A.I. Mees. An algorithm for non-linear network programming: Implementation, results and comparisons. Journal of Operational Research Society, 42:979–992, 1991a.Google Scholar
  24. N.L. Boland, C.J. Goh, and A.I. Mees. An algorithm for solving quadratic network flow problems. Applied Mathematics Letters, 4:61–64, 1991b.MATHCrossRefMathSciNetGoogle Scholar
  25. M. Bonatti and A. Gaivoronski. Guaranteed approximation of Markov chains with applications to multiplexer engineering in ATM networks. Annals of Operations Research, 49:111–136, 1994.MATHCrossRefGoogle Scholar
  26. D. Braess. Über ein paradoxen der werkehrsplannung. Unternehmenforschung, 12: 256–268, 1968.MathSciNetGoogle Scholar
  27. B. Calvert, W. Solomon, and I. Ziedins. Braess’s paradox in a queueing network with state-dependent routing. Journal of Applied Prob., 34:134–154, 1997.MATHCrossRefMathSciNetGoogle Scholar
  28. M.D. Canon and C.D. Cullum. A tight upper bound on the rate of convergence of the Frank-Wolfe algorithm. SIAM J. Control, 6:509–516, 1968.MATHCrossRefGoogle Scholar
  29. D.C. Cantor and M. Gerla. Optimal routing in a packet-switched computer network. IEEE Transactions on Computers, C-23:1062–1069, 1974.CrossRefMathSciNetGoogle Scholar
  30. J.-H. Chang and L. Tassiulas. Energy conserving routing in wireless ad-hoc networks. Technical report, Dept of Electrical and Computer Engineering, University of Maryland, College Park, MD 20742, 1997.Google Scholar
  31. J.E. Cohen and C. Jeffries. Congestion resulting from increased capacity in single-server queueing networks. IEEE/ACM Transactions on Networking, 5:1220–1225, 1997.CrossRefGoogle Scholar
  32. S.C. Dafermos and A. Nagurney. On some traffic equilibrium theory paradoxes. Transportation Research, 18B:101–110, 1984.MathSciNetGoogle Scholar
  33. S.C. Dafermos and F.T. Sparrow. The traffic assignment problem for a general network. Journal of RNBS, 73B:91–118, 1969.MathSciNetGoogle Scholar
  34. C.F. Daganzo. On the traffic assignment problem with flow dependent costs — I. Transportation Research, 11:433–437, 1977a.CrossRefGoogle Scholar
  35. C.F. Daganzo. On the traffic assignment problem with flow dependent costs — II. Transportation Research, 11:439–441, 1977b.CrossRefGoogle Scholar
  36. O. Damberg and A. Migdalas. Distributed disaggregate simplicial decomposition — A parallel algorithm for traffic assignment. In D. Hearn et al., editor, Network optimization, number 450 in Lecture Notes in Economics and Mathematical Systems, pages 172–193. Springer-Verlag, 1997a.Google Scholar
  37. O. Damberg and A. Migdalas. Parallel algorithms for network problems. In A. Migdalas, P.M. Pardalos, and S. Storøy, editors, Parallel computing in optimization, pages 183–238. Kluwer Academic Publishers, Dordrecht, 1997b.Google Scholar
  38. J.C. Dunn. Rate of convergence of conditional gradient algorithms near singular and nonsingular extremals. SIAM Journal on Control and Optimization, 17:187–211, 1979.MATHCrossRefMathSciNetGoogle Scholar
  39. S.P Evans. Derivation and analysis of some models for combining trip distribution and assignment. Transportation Research, 10:35–57, 1976.CrossRefGoogle Scholar
  40. M. Frank and P. Wolfe. An algorithm for quadratic programming. Naval Research Logistics Quarterly, 3:95–110, 1956.CrossRefMathSciNetGoogle Scholar
  41. L. Fratta, M. Gerla, and L. Kleinrock. The flow deviation method. An approach to store-and-forward communication network design. Networks, 3:97–133, 1973.MATHCrossRefMathSciNetGoogle Scholar
  42. S.A. Gabriel and D. Bernstein. The traffic equilibrium problem with nonadditive costs. Transportation Science, 31:337–348, 1997.MATHCrossRefGoogle Scholar
  43. S.A. Gabriel and D. Bernstein. Nonadditive shortest paths. Technical report, New Jersey TIDE Center, New Jersey Institute of Technology, Newark, NJ, 1999.Google Scholar
  44. N.H. Gartner. Analysis and control of transportation networks by Frank-Wolfe decomposition. In T Sasaki and T. Yamaoka, editors, Proceedings of the 7th International Symposium on Transportation and Traffic Flow, pages 591–623, Tokyo, 1977.Google Scholar
  45. N.H. Gartner. Optimal traffic assignment with elastic demands: A review. Part I: Analysis framework, Part II: Algorithmic approaches. Transportation Science, 14: 174–208, 1980.CrossRefMathSciNetGoogle Scholar
  46. A.M. Geoffrion. Elements of large-scale mathematical programming. Management Science, 16:652–691, 1970.CrossRefMathSciNetGoogle Scholar
  47. A.M. Geoffrion. Generalized Benders decomposition. Journal of Optimization Theory and Applications, 10:237–260, 1972.MATHCrossRefMathSciNetGoogle Scholar
  48. M. Gerla. Routing and flow control. In F.F. Kuo, editor, Protocols and techniques for data communication networks, chapter 4, pages 122–173. Prentice-Hall, 1981.Google Scholar
  49. M. Gerla and L. Kleinrock. Topological design of distributed computer networks. IEEE Trans. Comm., COM-25:48–60, 1977.CrossRefMathSciNetGoogle Scholar
  50. C.R. Glassey. A quadratic network optimization model for equilibrium single commodity trade flows. Mathematical Programming, 14:98–107, 1978.MATHCrossRefMathSciNetGoogle Scholar
  51. P. Gupta and P.R. Kumar. A system and traffic dependent adaptive routing algorithm for ad hoc networks. In Proceedings of the 36th IEEE Conference on Decision and Control, pages 2375–2380, San Diego, 1997.Google Scholar
  52. S.L. Hakimi. Optimum distribution of switching centers in a communication network and some related graph theoretic problems. Operations Research, 13:462–475, 1965.MATHCrossRefMathSciNetGoogle Scholar
  53. D.W. Hearn. The gap function of a convex program. Operations Research Letter, 1: 67–71, 1982.MATHCrossRefMathSciNetGoogle Scholar
  54. D.W. Hearn, S. Lawphongpanich, and J.A. Ventura. Finiteness in restricted simplicial decomposition. Operations Research Letters, 4:125–130, 1985.MATHCrossRefMathSciNetGoogle Scholar
  55. D.W. Hearn, S. Lawphongpanich, and J.A. Ventura. Restricted simplicial decomposition: Computation and extensions. Mathematical Programming Study, 31:99–118, 1987.MATHMathSciNetGoogle Scholar
  56. D.W. Hearn and J. Ribeira. Convergence of the Frank-Wolfe method for certain bounded variable traffic assignment problems. Transportation Research, 15B:437–442, 1981.Google Scholar
  57. M.P. Helme and T.L. Magnanti. Designing satellite communication networks by zero-one quadratic programming. Networks, 19:427–450, 1989.MATHCrossRefMathSciNetGoogle Scholar
  58. C.A. Holloway. An extension of the Frank and Wolfe method of feasible directions. Mathematical Programming, 6:14–27, 1974.MATHCrossRefMathSciNetGoogle Scholar
  59. B. Yaged Jr. Minimum cost routing for static network models. Networks, 1:139–172, 1971.MATHCrossRefMathSciNetGoogle Scholar
  60. H. Kameda, E. Altman, T. Kozawa, and Y. Hosokawa. Braess-like paradoxes in distributed computer systems, 2000. Submitted to IEEE Transactions on Automatic Control.Google Scholar
  61. A. Karakitsiou, A. Mavrommati, and A. Migdalas. Efficient minimization over products of simplices and its application to nonlinear multicommodity network problems. Operational Research International Journal, 4(2):99–118, 2005.CrossRefGoogle Scholar
  62. F.P. Kelly. Charging and rate control for elastic traffic. European Transactions on Telecommunications, 8:33–37, 1997.CrossRefGoogle Scholar
  63. L. Kleinrock. Communication nets: Stochastic message flow and delay. McGraw-Hill, New York, 1964.Google Scholar
  64. L. Kleinrock. Queueing systems, Volume II: Computer applications. John Wiley and Sons, New York, 1974.Google Scholar
  65. J.G. Klincewicz. Newton method for convex separable network flow problems. Networks, 13:427–442, 1983.MATHCrossRefMathSciNetGoogle Scholar
  66. W. Knödel. Graphentheoretische Methoden und ihre Anwendungen. Springer-Verlag, Berlin, 1969.MATHGoogle Scholar
  67. Y.A. Korillis, A.A. Lazar, and A. Orda. Capacity allocation under non-cooperative routing. IEEE/ACM Transactions on Networking, 5:309–173, 1997.CrossRefGoogle Scholar
  68. Y.A. Korillis, A.A. Lazar, and A. Orda. Avoiding the Braess paradox in non-cooperative networks. Journal of Applied Probability, 36:211–222, 1999.CrossRefMathSciNetGoogle Scholar
  69. M. Kourgiantakis, I. Mandalianos, A. Migdalas, and P. Pardalos. Optimization in e-commerce. In P.M. Pardalos and M.G.C. Resende, editors, Handbook of Optimization in Telecommunications. Springer, 2005. In this volume.Google Scholar
  70. T. Larsson and A. Migdalas. An algorithm for nonlinear programs over Cartesian product sets. Optimization, 21:535–542, 1990.MATHCrossRefMathSciNetGoogle Scholar
  71. T. Larsson, A. Migdalas, and M. Patriksson. The application of partial linearization algorithm to the traffic assignment problem. Optimization, 28:47–61, 1993.MATHCrossRefMathSciNetGoogle Scholar
  72. T. Larsson and M. Patriksson. Simplicial decomposition with disaggregated representation for traffic assignment problem. Transportation Science, 26:445–462, 1992.CrossRefGoogle Scholar
  73. T. Larsson, M. Patriksson, and C. Rydergren. Applications of the simplicial decomposition with nonlinear column generation to nonlinear network flows. In D. Heam et al., editor, Network optimization, volume 450 of Lecture Notes in Economics and Mathematical Systems, pages 346–373. Springer-Verlag, 1997.Google Scholar
  74. L.J. Leblanc, R.V. Helgason, and D.E. Boyce. Improved efficiency of the Frank-Wolfe algorithm for convex network. Transportation Science, 19:445–462, 1985.MATHCrossRefMathSciNetGoogle Scholar
  75. L.J. Leblanc, E.K. Morlok, and W.P. Pierskalla. An efficient approach to solving the road network equilibrium traffic assignment problem. Transportation Research, 9: 308–318, 1975.CrossRefGoogle Scholar
  76. T. Leventhal, G. Nemhauser, and L. Trotter Jr. A column generation algorithm for optimal traffic assignment. Transportation Science, 7:168–176, 1973.CrossRefMathSciNetGoogle Scholar
  77. E.S. Levitin and B.T. Polyak. Constrained minimization methods. USSR Computational Mathematics and Mathematical Physics, 6:1–50, 1966.CrossRefGoogle Scholar
  78. D.G. Luenberger. Linear and nonlinear programming. Addison-Wesley, Reading, Mass., second edition, 1984.MATHGoogle Scholar
  79. B. Martos. Nonlinear programming: Theory and methods. North-Holland, Amsterdam, 1975.MATHGoogle Scholar
  80. A. Migdalas. A regularization of the Frank-Wolfe method and unification of certain nonlinear programming methods. Mathematical Programming, 65:331–345, 1994.CrossRefMathSciNetGoogle Scholar
  81. A. Migdalas. Bilevel programming in traffic planning: Models, methods and challenge. Journal of Global Optimization, 7:381–405, 1995.MATHCrossRefMathSciNetGoogle Scholar
  82. A. Migdalas. Cyclic linearization and decomposition of team game models. In S. Butenko, R. Murphey, and P. Pardalos, editors, Recent developments in cooperative control and optimization, pages 333–348. Kluwer Academic Publishers, Boston, 2004.Google Scholar
  83. A. Migdalas, P.M. Pardalos, and P. Värbrand, editors. Multilevel optimization — Algorithms and applications. Kluwer Academic Publishers, 1997.Google Scholar
  84. M. Minoux. Subgradient optimization and Benders decomposition for large scale programming. In R.W. Cottle et al., editor, Mathematical programming, pages 271–288. North-Holland, Amsterdam, 1984.Google Scholar
  85. J.D. Murchland. Braess’s paradox of traffic flow. Transportation Research, 4:391–394, 1970.CrossRefGoogle Scholar
  86. S. Nguyen. An algorithm for the traffic assignment problem. Transportation Science, 8:203–216, 1974.CrossRefGoogle Scholar
  87. M.E. O’Kelly, D. Skorin-Kapov, and J. Skorin-Kapov. Lower bounds for the hub location problem. Management Science, 41:713–721, 1995.MATHCrossRefGoogle Scholar
  88. M. Patriksson. The traffic assignment problem: Models and methods. VSP, Utrecht, 1994.Google Scholar
  89. M. Patriksson. Parallel cost approximation algorithms for differentiable optimization. In A. Migdalas, P.M. Pardalos, and S. Storøy, editors, Parallel computing in optimization, pages 295–341. Kluwer Academic Publishers, Dordrecht, 1997.Google Scholar
  90. E.R. Petersen. A primal-dual traffic assignment algorithm. Management Science, 22: 87–95, 1975.MATHCrossRefMathSciNetGoogle Scholar
  91. M. De Prycker. Asynchronous transfer mode solution for broadband ISDN. Prentice-Hall, Englewood Cliffs, NJ, 1995.Google Scholar
  92. B.N. Pshenichny and Y.M. Danilin. Numerical methods in extremal problems. Mir Publishers, Moscow, 1978.Google Scholar
  93. J.B. Rosen. The gradient projection method for nonlinear programming. Part I: Linear constraints. SIAM Journal on Applied Mathematics, 8:181–217, 1960.MATHCrossRefGoogle Scholar
  94. T. Roughgarden. Selfish routing. PhD thesis, Cornell University, 2002.Google Scholar
  95. T. Roughgarden and E. Tardos. How bad is selfish routing? Journal of the ACM, 49: 239–259, 2002.CrossRefMathSciNetGoogle Scholar
  96. M. Schwartz. Computer-communication network design and analysis. Prentice-Hall, Englewood Cliffs, NJ, 1977.MATHGoogle Scholar
  97. M. Schwartz and C.K. Cheung. The gradient projection algorithm for multiple routing in message-switched networks. IEEE Trans. Comm., COM-24:449–456, 1976.CrossRefGoogle Scholar
  98. J. Seidler. Principles of computer communication network design. Ellis Horwood Ltd, Chichester, 1983.Google Scholar
  99. J.F. Shapiro. Mathematical programming — Structures and algorithms. John Wiley and Sons, New York, 1979.MATHGoogle Scholar
  100. PA. Steenbrink. Optimization of transport networks. John Wiley and Sons, London, 1974.Google Scholar
  101. B. von Hohenbalken. Simplicial decomposition in nonlinear programming algorithms. Mathematical Programming, 13:49–68, 1977.MATHCrossRefMathSciNetGoogle Scholar
  102. J.G. Wardrop. Some theoretical aspects of road traffic research. In Proceedings of the Institute of Civil Engineers — Part II, pages 325–378, 1952.Google Scholar
  103. A. Weintraub, C. Ortiz, and J. Conzalez. Accelerating convergence of the Frank-Wolfe algorithm. Transportation Research, 19B:113–122, 1985.Google Scholar
  104. P. Wolfe. Convergence theory in nonlinear programming. In J. Abadie, editor, Integer and nonlinear programming, pages 1–36. North-Holland, Amsterdam, 1970.Google Scholar
  105. H. Yaiche, R. Mazumdar, and C. Rosenberg. A game theoretic framework for bandwidth allocation and pricing of elastic connections in broadband networks: Theory and algorithms. IEEE/ACM Transaction on Networking, 8:667–678, 2000.CrossRefGoogle Scholar
  106. N. Zadeh. On building minimum cost communication networks. Networks, 3:315–331, 1973.MATHCrossRefMathSciNetGoogle Scholar
  107. W.I. Zangwill. The convex simplex method. Management Science, 14:221–283, 1967.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • Athanasios Migdalas
    • 1
  1. 1.DSS Laboratory ERGASYA Production Engineering & Management DepartmentTechnical University of CreteChaniaGreece

Personalised recommendations