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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Bibliography
E. Altman, R. El Azouzi, and V. Abramov. Non-cooperative routing in loss networks. Performance Evaluation, 49:43–55, 2002.
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.
M.S. Bazaraa and C.M. Shetty. Nonlinear programming — Theory and algorithms. John Wiley and Sons, New York, 1979.
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.
M. Beckmann, C.B. McGuire, and C.B. Winsten. Studies in Economics of Transportation. Yale University Press, 1956.
K.P. Bennet. Global tree optimization: A non-greedy decision tree algorithm. Computing Science and Statistics, 26:156–160, 1994.
K.P. Bennet and O.L. Mangasarian. Robust linear programming discrimination of two linearly inseparable sets. Optimization Methods and Software, 1:23–34, 1992.
K.P. Bennet and O.L. Mangasarian. Bilinear separation of two sets in n-space. Computational Optimization and Applications, 2:207–227, 1993.
J. Berechman. Highway-capacity utilization and investment in transportation corridors. Environment and Planning, 16A: 1475–1488, 1984.
F. Berggren. Power control and adaptive resource allocation in DS-CDMA systems. PhD thesis, Royal Institute of Technology, 2003.
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.
D.P. Bertsekas. On the Goldstein-Levitin-Polyak gradient projection method. IEEE Trans. Automat. Control, AC-21:174–184, 1976.
D.P. Bertsekas. Constrained optimization and Lagrange multiplier methods. Academic Press, New York, 1982a.
D.P. Bertsekas. Projected Newton methods for optimization problems with simple constraints. SIAM Journal on Control and Optimization, 20:221–246, 1982b.
D.P. Bertsekas and E.M. Gafni. Projected Newton methods and optimization of multicommodity flows. IEEE Trans. Automat. Control, AC-28:1090–1096, 1983.
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.
D.P. Bertsekas and R. Gallager. Data networks. Prentice-Hall, Englewood Cliffs, NJ, 1992.
D.P. Bertsekas and J.N. Tsitsiklis. Parallel and distributed computation — Numerical methods. Prentice-Hall, 1989.
D. Bienstock and O. Raskina. Aymptotic analysis of the flow deviation method for the maximum concurrent flow problem. Mathematical Programming, 91:479–492, 2002.
D. Bienstock and I. Saniee. ATM network design: Traffic models and optimization-based heuristics. Technical Report 98-20, DIMACS, 1998.
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.
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.
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.
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.
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.
D. Braess. Über ein paradoxen der werkehrsplannung. Unternehmenforschung, 12: 256–268, 1968.
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.
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.
D.C. Cantor and M. Gerla. Optimal routing in a packet-switched computer network. IEEE Transactions on Computers, C-23:1062–1069, 1974.
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.
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.
S.C. Dafermos and A. Nagurney. On some traffic equilibrium theory paradoxes. Transportation Research, 18B:101–110, 1984.
S.C. Dafermos and F.T. Sparrow. The traffic assignment problem for a general network. Journal of RNBS, 73B:91–118, 1969.
C.F. Daganzo. On the traffic assignment problem with flow dependent costs — I. Transportation Research, 11:433–437, 1977a.
C.F. Daganzo. On the traffic assignment problem with flow dependent costs — II. Transportation Research, 11:439–441, 1977b.
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.
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.
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.
S.P Evans. Derivation and analysis of some models for combining trip distribution and assignment. Transportation Research, 10:35–57, 1976.
M. Frank and P. Wolfe. An algorithm for quadratic programming. Naval Research Logistics Quarterly, 3:95–110, 1956.
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.
S.A. Gabriel and D. Bernstein. The traffic equilibrium problem with nonadditive costs. Transportation Science, 31:337–348, 1997.
S.A. Gabriel and D. Bernstein. Nonadditive shortest paths. Technical report, New Jersey TIDE Center, New Jersey Institute of Technology, Newark, NJ, 1999.
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.
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.
A.M. Geoffrion. Elements of large-scale mathematical programming. Management Science, 16:652–691, 1970.
A.M. Geoffrion. Generalized Benders decomposition. Journal of Optimization Theory and Applications, 10:237–260, 1972.
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.
M. Gerla and L. Kleinrock. Topological design of distributed computer networks. IEEE Trans. Comm., COM-25:48–60, 1977.
C.R. Glassey. A quadratic network optimization model for equilibrium single commodity trade flows. Mathematical Programming, 14:98–107, 1978.
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.
S.L. Hakimi. Optimum distribution of switching centers in a communication network and some related graph theoretic problems. Operations Research, 13:462–475, 1965.
D.W. Hearn. The gap function of a convex program. Operations Research Letter, 1: 67–71, 1982.
D.W. Hearn, S. Lawphongpanich, and J.A. Ventura. Finiteness in restricted simplicial decomposition. Operations Research Letters, 4:125–130, 1985.
D.W. Hearn, S. Lawphongpanich, and J.A. Ventura. Restricted simplicial decomposition: Computation and extensions. Mathematical Programming Study, 31:99–118, 1987.
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.
M.P. Helme and T.L. Magnanti. Designing satellite communication networks by zero-one quadratic programming. Networks, 19:427–450, 1989.
C.A. Holloway. An extension of the Frank and Wolfe method of feasible directions. Mathematical Programming, 6:14–27, 1974.
B. Yaged Jr. Minimum cost routing for static network models. Networks, 1:139–172, 1971.
H. Kameda, E. Altman, T. Kozawa, and Y. Hosokawa. Braess-like paradoxes in distributed computer systems, 2000. Submitted to IEEE Transactions on Automatic Control.
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.
F.P. Kelly. Charging and rate control for elastic traffic. European Transactions on Telecommunications, 8:33–37, 1997.
L. Kleinrock. Communication nets: Stochastic message flow and delay. McGraw-Hill, New York, 1964.
L. Kleinrock. Queueing systems, Volume II: Computer applications. John Wiley and Sons, New York, 1974.
J.G. Klincewicz. Newton method for convex separable network flow problems. Networks, 13:427–442, 1983.
W. Knödel. Graphentheoretische Methoden und ihre Anwendungen. Springer-Verlag, Berlin, 1969.
Y.A. Korillis, A.A. Lazar, and A. Orda. Capacity allocation under non-cooperative routing. IEEE/ACM Transactions on Networking, 5:309–173, 1997.
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.
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.
T. Larsson and A. Migdalas. An algorithm for nonlinear programs over Cartesian product sets. Optimization, 21:535–542, 1990.
T. Larsson, A. Migdalas, and M. Patriksson. The application of partial linearization algorithm to the traffic assignment problem. Optimization, 28:47–61, 1993.
T. Larsson and M. Patriksson. Simplicial decomposition with disaggregated representation for traffic assignment problem. Transportation Science, 26:445–462, 1992.
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.
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.
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.
T. Leventhal, G. Nemhauser, and L. Trotter Jr. A column generation algorithm for optimal traffic assignment. Transportation Science, 7:168–176, 1973.
E.S. Levitin and B.T. Polyak. Constrained minimization methods. USSR Computational Mathematics and Mathematical Physics, 6:1–50, 1966.
D.G. Luenberger. Linear and nonlinear programming. Addison-Wesley, Reading, Mass., second edition, 1984.
B. Martos. Nonlinear programming: Theory and methods. North-Holland, Amsterdam, 1975.
A. Migdalas. A regularization of the Frank-Wolfe method and unification of certain nonlinear programming methods. Mathematical Programming, 65:331–345, 1994.
A. Migdalas. Bilevel programming in traffic planning: Models, methods and challenge. Journal of Global Optimization, 7:381–405, 1995.
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.
A. Migdalas, P.M. Pardalos, and P. Värbrand, editors. Multilevel optimization — Algorithms and applications. Kluwer Academic Publishers, 1997.
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.
J.D. Murchland. Braess’s paradox of traffic flow. Transportation Research, 4:391–394, 1970.
S. Nguyen. An algorithm for the traffic assignment problem. Transportation Science, 8:203–216, 1974.
M.E. O’Kelly, D. Skorin-Kapov, and J. Skorin-Kapov. Lower bounds for the hub location problem. Management Science, 41:713–721, 1995.
M. Patriksson. The traffic assignment problem: Models and methods. VSP, Utrecht, 1994.
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.
E.R. Petersen. A primal-dual traffic assignment algorithm. Management Science, 22: 87–95, 1975.
M. De Prycker. Asynchronous transfer mode solution for broadband ISDN. Prentice-Hall, Englewood Cliffs, NJ, 1995.
B.N. Pshenichny and Y.M. Danilin. Numerical methods in extremal problems. Mir Publishers, Moscow, 1978.
J.B. Rosen. The gradient projection method for nonlinear programming. Part I: Linear constraints. SIAM Journal on Applied Mathematics, 8:181–217, 1960.
T. Roughgarden. Selfish routing. PhD thesis, Cornell University, 2002.
T. Roughgarden and E. Tardos. How bad is selfish routing? Journal of the ACM, 49: 239–259, 2002.
M. Schwartz. Computer-communication network design and analysis. Prentice-Hall, Englewood Cliffs, NJ, 1977.
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.
J. Seidler. Principles of computer communication network design. Ellis Horwood Ltd, Chichester, 1983.
J.F. Shapiro. Mathematical programming — Structures and algorithms. John Wiley and Sons, New York, 1979.
PA. Steenbrink. Optimization of transport networks. John Wiley and Sons, London, 1974.
B. von Hohenbalken. Simplicial decomposition in nonlinear programming algorithms. Mathematical Programming, 13:49–68, 1977.
J.G. Wardrop. Some theoretical aspects of road traffic research. In Proceedings of the Institute of Civil Engineers — Part II, pages 325–378, 1952.
A. Weintraub, C. Ortiz, and J. Conzalez. Accelerating convergence of the Frank-Wolfe algorithm. Transportation Research, 19B:113–122, 1985.
P. Wolfe. Convergence theory in nonlinear programming. In J. Abadie, editor, Integer and nonlinear programming, pages 1–36. North-Holland, Amsterdam, 1970.
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.
N. Zadeh. On building minimum cost communication networks. Networks, 3:315–331, 1973.
W.I. Zangwill. The convex simplex method. Management Science, 14:221–283, 1967.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer Science+Business Media, Inc.
About this chapter
Cite this chapter
Migdalas, A. (2006). Nonlinear Programming in Telecommunications. In: Resende, M.G.C., Pardalos, P.M. (eds) Handbook of Optimization in Telecommunications. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-30165-5_2
Download citation
DOI: https://doi.org/10.1007/978-0-387-30165-5_2
Publisher Name: Springer, Boston, MA
Print ISBN: 978-0-387-30662-9
Online ISBN: 978-0-387-30165-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)