Projected Newton methods and optimization of multicommodity flows

A superlinearly convergent Newton-like method for linearly constrained optimization problems is adapted for solution of multicommodity network flow problems of the type arising in communication and transportation networks. We show that the method can be implemented approximately by making use of conjugate gradient iterations without the need to compute explicitly the Hessian matrix. Preliminary computational results suggest that this type of method is capable of yielding highly accurate solutions of nonlinear multicommodity flow problems far more efficiently than any of the methods available at present.

[1]  Philip Wolfe,et al.  An algorithm for quadratic programming , 1956 .

[2]  James M. Ortega,et al.  Iterative solution of nonlinear equations in several variables , 2014, Computer science and applied mathematics.

[3]  Michael Florian,et al.  A Method for Computing Network Equilibrium with Elastic Demands , 1974 .

[4]  D. Bertsekas.  Partial conjugate gradient methods for a class of optimal control problems , 1974 .

[5]  Olvi L. Mangasarian,et al.  Superlinearly convergent quasi-newton algorithms for nonlinearly constrained optimization problems , 1976, Math. Program..

[6]  Michael Florian.  An improved linear approximation algorithm for the network equilibrium (packet switching) problem , 1977, 1977 IEEE Conference on Decision and Control including the 16th Symposium on Adaptive Processes and A Special Symposium on Fuzzy Set Theory and Applications.

[7]  Eliezer M. Gafni,et al.  Convergence of a routing algorithm , 1979 .

[8]  Melanie L. Lenard,et al.  A computational study of active set strategies in nonlinear programming with linear constraints , 1979, Math. Program..

[9]  Dimitri P. Bertsekas,et al.  Algorithms for Nonlinear Multicommodity Network Flow Problems , 1979 .

[10]  J. Dunn.  Newton’s Method and the Goldstein Step-Length Rule for Constrained Minimization Problems , 1980 .

[11]  D. Bertsekas.  A class of optimal routing algorithms for communication networks , 1980 .

[12]  Stella Dafermos,et al.  Traffic Equilibrium and Variational Inequalities , 1980 .

[13]  Robert G. Gallager,et al.  Flow Control and Routing Algorithms for Data Networks , 1980 .

[14]  Magnus R. Hestenes,et al.  Conjugate Direction Methods in Optimization , 1980 .

[15]  J. Dunn.  Global and Asymptotic Convergence Rate Estimates for a Class of Projected Gradient Processes , 1981 .

[16]  D. Bertsekas.  Projected Newton methods for optimization problems with simple constraints , 1981, 1981 20th IEEE Conference on Decision and Control including the Symposium on Adaptive Processes.

[17]  T. Magnanti,et al.  Equilibria on a Congested Transportation Network , 1981 .

[18]  R. Dembo,et al.  A scaled reduced gradient algorithm for network flow problems with convex separable costs , 1981 .

[19]  R. Dembo,et al.  INEXACT NEWTON METHODS , 1982 .

[20]  D. Bertsekas,et al.  Projection methods for variational inequalities with application to the traffic assignment problem , 1982 .