Abstract
Based on computational experiments with different approaches to convex separable network flow problems a hybrid algorithm is developed and implemented. Phase one of the algorithm uses a rapidly converging series of piecewise linear secant approximations in order to determine a good solution within some distance of the optimum. Starting from this solution, a feasible direction method, based on reduced Newton directions, is used in the second phase of the algorithm to determine the optimal solution. Since nonlinear network flow problems tend to be degenerate, special emphasis is put on the construction of a basis that yields a strictly positive step length at the beginning of phase two of the hybrid algorithm.
A number of test problems have been solved successfully. It is expected that the approach can be extended to solve large-scale problems with convex separable objective functions. Details of the implementation and computational results are presented.
Zusammenfassung
Ausgehend von experimentellen Ergebnissen mit unterschiedlichen Lösungsverfahren für separable Netzwerkflußprobleme wurde ein zweistufiges Verfahren entwickelt und implementiert. Auf der ersten Stufe wird in einem iterativen Prozeß das zu lösende Problem mehrfach stückweise linearisiert. Man erhält eine bereits sehr gute Lösung. Mit dieser wird ein Richtungsverfahren initialisiert, das unter Verwendung reduzierter Newton Richtungen die optimale Lösung bestimmt. Das Richtungsverfahren bildet die zweite Stufe des Verfahrens. Da nichtlineare Netzwerkflußprobleme im allgemeinen stark entartet sind, wird zu Beginn der zweiten Stufe des beschriebenen Verfahrens eine Basis konstruiert, die eine positive Schrittlänge zuläßt.
Es wurden zahlreiche Testprobleme mit bis zu 600 Knoten und 1400 Kanten mit dem beschriebenen Verfahren erfolgreich gelöst. Es wird erwartet, daß das Verfahren auch auf sehr viel größere Probleme mit konvexer, separabler Zielfunktion angewendet werden kann. Es wird auf Fragen zur Implementation eingegangen und es werden numerische Ergebnisse diskutiert.
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References
Ali, A., R. Helgason, J. Kennington, andH. Lall: Primal Simplex Network Codes: State-of-the-Art Implementation Technology. Networks8, 1978, 315–338.
Bazaraa, M.S., andC.M. Shetty: Nonlinear Programming — Theory and Algorithms. John Wiley & Sons, New York 1979.
Beale, E.M.L.: An Algorithm for Solving the Transportation Problem when the Shipping Cost over each Route is Convex. Naval Research Logistics Quarterly6, 1959, 43–56.
Bland, R.G.: New Finite Pivoting Rules for the Simplex Method. Mathematics of Operations Research2, 1977, 103–107.
Bradley, G.H., G.G. Brown, andG.W. Graves: Design and Implementation of Large Scale Primal Transshipment Algorithms. Management Science24, 1977, 1–34.
Belling-Seib, K.: Mathematische Programmierungsprobleme mit Netzwerkstruktur unter besonderer Berücksichtigung von Netzwerkflußproblemen mit konvexen, separablen Kostenverläufen. Dissertation am Fachbereich Wirtschaftswissenschaften der Freien Universität Berlin 1983.
—: Methods for Solving Nonlinear Network Flow Problems. Methods of Operations Research51, 1984, 227–238.
Charnes, A., andW.W. Cooper: Nonlinear Network Flows and Convex Programming over Incidence Matrices. Naval Research Logistics Quarterly5, 1958, 231–240.
Collins, M., L. Cooper, R. Helgason, J. Kennington, andL. LeBlanc: Solving the Pipe Network Analysis Problem Using Optimization Techniques. Management Science24, 1978, 747–760.
Cooper, L., andJ. Kennington: Steady-State Analysis of Nonlinear Resistive Electrical Networks using Optimization Techniques. Technical Report IEOR 77012, Southern Methodist University Dallas, TX, 1977.
Cottle, R., andJ. Krarup, ed.: Optimization Methods for Resource Allocation. The English Universities Press ltd, London 1974.
Dafermos, St. C.: An Extended Traffic Assignment Model with Applications to Two-Way Traffic. Transportation Science5, 1971, 366–389.
Dembo, R.S., andJ.G. Klincewicz: A Scaled Reduced Gradient Algorithm for Network Flow with Convex Separable Costs. Mathematical Programming Study15, 1981, 125–147.
Fletcher, R.: Practical Methods of Optimization. Volume 1: Unconstrained Optimization. Volume 2: Constrained Optimization. John Wiley & Sons, Chichester 1980/81.
Florian, M.: A Traffic Equilibrium Model of Travel by Car and Public Transportation Modes. Transportation Science11, 1977, 166–179.
Florian, M., andS. Nguyen: An Application and Validation of Equilibrium Trip Assignment Methods. Transportation Science10, 1976, 374–390.
Gill, Ph.E., andW. Murray: Numerical Methods for Constrained Optimization. Academic Press, London 1974.
Gill, Ph. E., W. Murray, M.A. Saunders, andM.H. Wright: A Note on a Sufficient-Decrease Criterion for a Non-Derivative Step-Length Procedure. Mathematical Programming23, 1982, 349–352.
Glover, F., D. Karney, andD. Klingman: Implementation and Computational Comparisons of Primal, Dual, and Primal-Dual Computer Codes for Minimum Cost Network Flow Problems. Networks4, 1974, 191–212.
Helgason, R., A. Ali, andJ. Kennington: The Convex Cost Network Flow Problem: A Survey of Algorithms. Technical Report OREM 78001, Southern Methodist University, Dallas, TX 1978.
Horowitz, E., andS. Sahni: Fundamentals of Computer Algorithms. Pitman Publishing ltd, London 1978.
Hu, T.C.: Minimum Cost Flows in Convex Cost Networks. Naval Research Logistics Quarterly13, 1966, 1–9.
Irwin, C.L., andCh.W. Yang: Iteration and Sensitivity for a Spatial Equilibrium Problem with Linear Supply and Demand Functions. Operations Research30, 1982, 319–335.
Klein, M.: The Primal Method for Minimal Cost Flows with Applications to the Assignment and Transportation Problem. Management Science14, 1967, 205–220.
Klingman, D., A. Napier, andJ. Stutz: NETGEN, A Program for Generating Large Scale Capacitated Assignment, Transportation, and Minimum Cost Flow Network Problems. Management Science20, 1974, 814–821.
Klingman, D., andD. Karney: Implementation and Computational Study on an In-Core, Out-of-Core Primal Network Code. Operations Research24, 1975, 1056–1077.
LeBlanc, L., E. Morlok, andW. Pierskalla: An Efficient Approach to Solving Road Network Equilibrium Traffic Assignment Problems. Transportation Research9, 1975, 309–318.
Leventhal, T., G. Nemhauser, andL. Trotter jr.: A Column Generation Algorithm for Optimal Traffic Assignment. Transportation Science7, 1973, 168–176.
Luenberger, D.G.: Introduction ot Linear and Nonlinear Programming. Addison-Wesley, Reading, MA, 1973.
Magnanri, T.L., andB.L. Golden: Transportation Planning: Network Models and their Implementation. In: Studies in Operations Management; A.C. Hax ed., North-Holland Publ., Amsterdam 1978, 465–517.
Meyer, R.R., andC.Y. Kao: Secant Approximation Methods for Convex Optimization. Mathematical Programming Study14, 1981, 143–162.
Murtagh, B.A.: Advanced Linear Programming: Computation and Practice. McGraw-Hill Inc., 1981.
Murtagh, B.A., andM.A. Saunders: Large-Scale Linearly Constrained Optimization. Mathematical Programming14, 1978, 41–72.
Nguyen, S.: Une Approche Unifiée des Methodes d'Equilibre pour l'Affectation du Traffic. Publication no. 171, Départment d'Informatique, Université de Montréal, Montréal, Quebec, 1974.
Ortega, J.M., andW.C. Rheinboldt: Iterative Methods of Nonlinear Equations in Severable Variables. Academic Press, New York 1970.
Pang, J.-Sh., andCH-S. Yu: A Special Spatial Equilibrium Problem. Networks14, 1984, 75–81.
Rosenthal, R.E.: A Nonlinear Network Flow Algorithm for Maximization of Benefits in a Hydro-electronic Power System. Operations Research29, 1981, 763–786.
Samuelson, P.A.: Spatial Price Equilibrium and Linear Programming. The American Economic Review42, 1952, 283–303.
Takayama, T., andG.G. Judge: Spatial and Temporal Price and Allocation Models. North-Holland Publ., Amsterdam 1971.
Weintraub, A., andJ. González: An Algorithm for the Traffic Assignment Problem. Networks10, 1980, 197–209.
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Belling-Seib, K. A hybrid algorithm for solving convex separable network flow problems. Zeitschrift für Operations Research 29, 105–123 (1985). https://doi.org/10.1007/BF01918199
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DOI: https://doi.org/10.1007/BF01918199