# Nonlinear Programming in Telecommunications

Chapter

## Abstract

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.

## Keywords

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## Preview

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## Bibliography

- E. Altman, R. El Azouzi, and V. Abramov. Non-cooperative routing in loss networks.
*Performance Evaluation*, 49:43–55, 2002.MATHCrossRefGoogle Scholar - 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
- M.S. Bazaraa and C.M. Shetty.
*Nonlinear programming — Theory and algorithms*. John Wiley and Sons, New York, 1979.MATHGoogle Scholar - 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 - M. Beckmann, C.B. McGuire, and C.B. Winsten.
*Studies in Economics of Transportation*. Yale University Press, 1956.Google Scholar - K.P. Bennet. Global tree optimization: A non-greedy decision tree algorithm.
*Computing Science and Statistics*, 26:156–160, 1994.Google Scholar - 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 - 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 - J. Berechman. Highway-capacity utilization and investment in transportation corridors.
*Environment and Planning*, 16A: 1475–1488, 1984.CrossRefGoogle Scholar - F. Berggren.
*Power control and adaptive resource allocation in DS-CDMA systems*. PhD thesis, Royal Institute of Technology, 2003.Google Scholar - 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
- D.P. Bertsekas. On the Goldstein-Levitin-Polyak gradient projection method.
*IEEE Trans. Automat. Control*, AC-21:174–184, 1976.CrossRefMathSciNetGoogle Scholar - D.P. Bertsekas.
*Constrained optimization and Lagrange multiplier methods*. Academic Press, New York, 1982a.MATHGoogle Scholar - D.P. Bertsekas. Projected Newton methods for optimization problems with simple constraints.
*SIAM Journal on Control and Optimization*, 20:221–246, 1982b.MATHCrossRefMathSciNetGoogle Scholar - 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 - 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 - D.P. Bertsekas and R. Gallager.
*Data networks*. Prentice-Hall, Englewood Cliffs, NJ, 1992.MATHGoogle Scholar - D.P. Bertsekas and J.N. Tsitsiklis.
*Parallel and distributed computation — Numerical methods*. Prentice-Hall, 1989.Google Scholar - 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 - D. Bienstock and I. Saniee. ATM network design: Traffic models and optimization-based heuristics. Technical Report 98-20, DIMACS, 1998.Google Scholar
- 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
- 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 - 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 - 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 - 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 - D. Braess. Über ein paradoxen der werkehrsplannung.
*Unternehmenforschung*, 12: 256–268, 1968.MathSciNetGoogle Scholar - 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 - 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 - 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 - 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
- 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 - S.C. Dafermos and A. Nagurney. On some traffic equilibrium theory paradoxes.
*Transportation Research*, 18B:101–110, 1984.MathSciNetGoogle Scholar - S.C. Dafermos and F.T. Sparrow. The traffic assignment problem for a general network.
*Journal of RNBS*, 73B:91–118, 1969.MathSciNetGoogle Scholar - C.F. Daganzo. On the traffic assignment problem with flow dependent costs — I.
*Transportation Research*, 11:433–437, 1977a.CrossRefGoogle Scholar - C.F. Daganzo. On the traffic assignment problem with flow dependent costs — II.
*Transportation Research*, 11:439–441, 1977b.CrossRefGoogle Scholar - 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 - 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 - 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 - S.P Evans. Derivation and analysis of some models for combining trip distribution and assignment.
*Transportation Research*, 10:35–57, 1976.CrossRefGoogle Scholar - M. Frank and P. Wolfe. An algorithm for quadratic programming.
*Naval Research Logistics Quarterly*, 3:95–110, 1956.CrossRefMathSciNetGoogle Scholar - 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 - S.A. Gabriel and D. Bernstein. The traffic equilibrium problem with nonadditive costs.
*Transportation Science*, 31:337–348, 1997.MATHCrossRefGoogle Scholar - 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
- 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 - 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 - A.M. Geoffrion. Elements of large-scale mathematical programming.
*Management Science*, 16:652–691, 1970.CrossRefMathSciNetGoogle Scholar - A.M. Geoffrion. Generalized Benders decomposition.
*Journal of Optimization Theory and Applications*, 10:237–260, 1972.MATHCrossRefMathSciNetGoogle Scholar - 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 - M. Gerla and L. Kleinrock. Topological design of distributed computer networks.
*IEEE Trans. Comm.*, COM-25:48–60, 1977.CrossRefMathSciNetGoogle Scholar - C.R. Glassey. A quadratic network optimization model for equilibrium single commodity trade flows.
*Mathematical Programming*, 14:98–107, 1978.MATHCrossRefMathSciNetGoogle Scholar - 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 - 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 - D.W. Hearn. The gap function of a convex program.
*Operations Research Letter*, 1: 67–71, 1982.MATHCrossRefMathSciNetGoogle Scholar - D.W. Hearn, S. Lawphongpanich, and J.A. Ventura. Finiteness in restricted simplicial decomposition.
*Operations Research Letters*, 4:125–130, 1985.MATHCrossRefMathSciNetGoogle Scholar - D.W. Hearn, S. Lawphongpanich, and J.A. Ventura. Restricted simplicial decomposition: Computation and extensions.
*Mathematical Programming Study*, 31:99–118, 1987.MATHMathSciNetGoogle Scholar - 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 - M.P. Helme and T.L. Magnanti. Designing satellite communication networks by zero-one quadratic programming.
*Networks*, 19:427–450, 1989.MATHCrossRefMathSciNetGoogle Scholar - C.A. Holloway. An extension of the Frank and Wolfe method of feasible directions.
*Mathematical Programming*, 6:14–27, 1974.MATHCrossRefMathSciNetGoogle Scholar - B. Yaged Jr. Minimum cost routing for static network models.
*Networks*, 1:139–172, 1971.MATHCrossRefMathSciNetGoogle Scholar - 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
- 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 - F.P. Kelly. Charging and rate control for elastic traffic.
*European Transactions on Telecommunications*, 8:33–37, 1997.CrossRefGoogle Scholar - L. Kleinrock.
*Communication nets: Stochastic message flow and delay*. McGraw-Hill, New York, 1964.Google Scholar - L. Kleinrock.
*Queueing systems, Volume II: Computer applications*. John Wiley and Sons, New York, 1974.Google Scholar - J.G. Klincewicz. Newton method for convex separable network flow problems.
*Networks*, 13:427–442, 1983.MATHCrossRefMathSciNetGoogle Scholar - W. Knödel.
*Graphentheoretische Methoden und ihre Anwendungen*. Springer-Verlag, Berlin, 1969.MATHGoogle Scholar - 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 - 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 - 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 - T. Larsson and A. Migdalas. An algorithm for nonlinear programs over Cartesian product sets.
*Optimization*, 21:535–542, 1990.MATHCrossRefMathSciNetGoogle Scholar - 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 - T. Larsson and M. Patriksson. Simplicial decomposition with disaggregated representation for traffic assignment problem.
*Transportation Science*, 26:445–462, 1992.CrossRefGoogle Scholar - 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 - 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 - 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 - T. Leventhal, G. Nemhauser, and L. Trotter Jr. A column generation algorithm for optimal traffic assignment.
*Transportation Science*, 7:168–176, 1973.CrossRefMathSciNetGoogle Scholar - E.S. Levitin and B.T. Polyak. Constrained minimization methods.
*USSR Computational Mathematics and Mathematical Physics*, 6:1–50, 1966.CrossRefGoogle Scholar - D.G. Luenberger.
*Linear and nonlinear programming*. Addison-Wesley, Reading, Mass., second edition, 1984.MATHGoogle Scholar - B. Martos.
*Nonlinear programming: Theory and methods*. North-Holland, Amsterdam, 1975.MATHGoogle Scholar - A. Migdalas. A regularization of the Frank-Wolfe method and unification of certain nonlinear programming methods.
*Mathematical Programming*, 65:331–345, 1994.CrossRefMathSciNetGoogle Scholar - A. Migdalas. Bilevel programming in traffic planning: Models, methods and challenge.
*Journal of Global Optimization*, 7:381–405, 1995.MATHCrossRefMathSciNetGoogle Scholar - 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 - A. Migdalas, P.M. Pardalos, and P. Värbrand, editors.
*Multilevel optimization — Algorithms and applications*. Kluwer Academic Publishers, 1997.Google Scholar - 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 - J.D. Murchland. Braess’s paradox of traffic flow.
*Transportation Research*, 4:391–394, 1970.CrossRefGoogle Scholar - S. Nguyen. An algorithm for the traffic assignment problem.
*Transportation Science*, 8:203–216, 1974.CrossRefGoogle Scholar - 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 - M. Patriksson.
*The traffic assignment problem: Models and methods*. VSP, Utrecht, 1994.Google Scholar - 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 - E.R. Petersen. A primal-dual traffic assignment algorithm.
*Management Science*, 22: 87–95, 1975.MATHCrossRefMathSciNetGoogle Scholar - M. De Prycker.
*Asynchronous transfer mode solution for broadband ISDN*. Prentice-Hall, Englewood Cliffs, NJ, 1995.Google Scholar - B.N. Pshenichny and Y.M. Danilin.
*Numerical methods in extremal problems*. Mir Publishers, Moscow, 1978.Google Scholar - 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 - T. Roughgarden.
*Selfish routing*. PhD thesis, Cornell University, 2002.Google Scholar - T. Roughgarden and E. Tardos. How bad is selfish routing?
*Journal of the ACM*, 49: 239–259, 2002.CrossRefMathSciNetGoogle Scholar - M. Schwartz.
*Computer-communication network design and analysis*. Prentice-Hall, Englewood Cliffs, NJ, 1977.MATHGoogle Scholar - 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 - J. Seidler.
*Principles of computer communication network design*. Ellis Horwood Ltd, Chichester, 1983.Google Scholar - J.F. Shapiro.
*Mathematical programming — Structures and algorithms*. John Wiley and Sons, New York, 1979.MATHGoogle Scholar - PA. Steenbrink.
*Optimization of transport networks*. John Wiley and Sons, London, 1974.Google Scholar - B. von Hohenbalken. Simplicial decomposition in nonlinear programming algorithms.
*Mathematical Programming*, 13:49–68, 1977.MATHCrossRefMathSciNetGoogle Scholar - 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 - A. Weintraub, C. Ortiz, and J. Conzalez. Accelerating convergence of the Frank-Wolfe algorithm.
*Transportation Research*, 19B:113–122, 1985.Google Scholar - P. Wolfe. Convergence theory in nonlinear programming. In J. Abadie, editor,
*Integer and nonlinear programming*, pages 1–36. North-Holland, Amsterdam, 1970.Google Scholar - 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 - N. Zadeh. On building minimum cost communication networks.
*Networks*, 3:315–331, 1973.MATHCrossRefMathSciNetGoogle Scholar - W.I. Zangwill. The convex simplex method.
*Management Science*, 14:221–283, 1967.MATHCrossRefMathSciNetGoogle Scholar

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