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A hybrid algorithm for solving convex separable network flow problems

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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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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

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