Separable convexification and DCA techniques for capacity and flow assignment problems
RAIRO - Operations Research - Recherche Opérationnelle, Volume 35 (2001) no. 2, p. 269-281

We study a continuous version of the capacity and flow assignment problem (CFA) where the design cost is combined with an average delay measure to yield a non convex objective function coupled with multicommodity flow constraints. A separable convexification of each arc cost function is proposed to obtain approximate feasible solutions within easily computable gaps from optimality. On the other hand, DC (difference of convex functions) programming can be used to compute accurate upper bounds and reduce the gap. The technique is shown to be effective when topology is assumed fixed and capacity expansion on some arcs is considered.

On étudie ici une version continue du problème de dimensionnement et routage dans un réseau de communications, dans lequel les coûts de conception sont combinés aux mesures de délai moyen d'acheminement, engendrant un problème de multiflots avec une fonction objectif non convexe. On propose un encadrement de la valeur optimale par convexification séparable sur les arcs, d'une part, et par calcul d'optima locaux issus d'un modèle DC (différence de fonctions convexes) des fonctions de coût. Cette dernière technique permet de réduire la distance à la valeur optimale et on illustre son efficacité sur des problèmes d'expansion de capacités.

Keywords: network design, DC optimization, capacity and flow assignment
@article{RO_2001__35_2_269_0,
     author = {Mahey, P. and Phong, Thai Q. and Luna, H. P. L.},
     title = {Separable convexification and DCA techniques for capacity and flow assignment problems},
     journal = {RAIRO - Operations Research - Recherche Op\'erationnelle},
     publisher = {EDP-Sciences},
     volume = {35},
     number = {2},
     year = {2001},
     pages = {269-281},
     zbl = {1048.90047},
     mrnumber = {1868872},
     language = {en},
     url = {http://www.numdam.org/item/RO_2001__35_2_269_0}
}
Mahey, P.; Phong, Thai Q.; Luna, H. P. L. Separable convexification and DCA techniques for capacity and flow assignment problems. RAIRO - Operations Research - Recherche Opérationnelle, Volume 35 (2001) no. 2, pp. 269-281. http://www.numdam.org/item/RO_2001__35_2_269_0/

[1] A. Balakrishnan and S.C. Graves, A composite algorithm for a concave-cost network flow problem. Networks 19 (1989) 175-202. | MR 984565 | Zbl 0673.90034

[2] D.P. Bertsekas and R.G. Gallager, Data Networks. Prentice-Hall (1987). | Zbl 0734.68006

[3] J.E. Falk, Lagrange multipliers and nonconvex programs. SIAM J. Control Optim. 7 (1969) 534-545. | MR 270749 | Zbl 0184.44404

[4] L. Fratta, M. Gerla and L. Kleinrock, The flow deviation method: an approach to store-and-forward communication network design. Networks 3 (1973) 97-133. | MR 312994 | Zbl 1131.90321

[5] B. Gavish, Augmented Lagrangian based bounds for centralized network design. IEEE Trans. Comm. 33 (1985) 1247-1257.

[6] B. Gavish and K. Altinkemer, Backbone network design tools with economic tradeoffs. ORSA J. Comput. 2/3 (1990) 236-252. | Zbl 0755.90024

[7] B. Gavish and I. Neuman, A system for routing and capacity assignment in computer communication networks. IEEE Trans. Comm. 37 (1989) 360-366.

[8] M. Gerla, The Design of Store-and-forward Networks for Computer Communications. Ph.D. Thesis, UCLA (1973).

[9] M. Gerla and L. Kleinrock, On the topological design of distributed computer networks. IEEE Trans. Comm. 25 (1977) 48-60. | MR 449892

[10] M. Gerla, J.A.S. Monteiro and R. Pazos, Topology design and bandwith allocation in ATM nets. IEEE J. Selected Areas in Communications 7 (1989) 1253-1261.

[11] J.B. Hiriart-Urruty and C. Lemaréchal, Convex Analysis and Minimization Algorithms. Springer-Verlag (1993). | Zbl 0795.49002

[12] H. Konno, P.T. Thach and H. Tuy, Optimization on Low Rank Nonconvex Structures. Kluwer Academic Publishers, Dordrecht (1997). | MR 1480917 | Zbl 0879.90171

[13] H.P.L. Luna and P. Mahey, Bounds for global optimization of capacity expansion and flow assignment problems. Oper. Res. Lett. 26 (2000) 211-216. | MR 1784600 | Zbl 0960.90055

[14] P. Mahey, A. Benchakroun and F. Boyer, Capacity and flow assignment of data networks by generalized Benders decomposition. J. Global Optim. (to appear). | MR 1841079 | Zbl 1002.90082

[15] P. Mahey, A. Ouorou, L. Leblanc and J. Chifflet, A new proximal decomposition algorithm for routing in telecommunications networks. Networks 31 (1998) 227-238. | Zbl 1015.90020

[16] A. Ouorou, P. Mahey and J.P. Vial, A survey of algorithms for convex multicommodity flow problems. Management Sci. 46 (2000) 126-147.

[17] P.D. Tao and L.T.H. An, Convex analysis approach to dc programming: Theory, algorithms and applications. Acta Math. Vietnam. 22 (1997) 289-355. | MR 1479751 | Zbl 0895.90152

[18] N.T. Quang, Une approche dc en optimisation dans les réseaux. Algorithmes, codes et simulations numériques. Doct. Thesis, Univ. Rouen (1999).

[19] B. Sanso, M. Gendreau and F. Soumis, An algorithm for network dimensioning under reliability considerations. Ann. Oper. Res. 36 (1992) 263-274. | Zbl 0825.90377

[20] H. Tuy, S. Ghannadan, A. Migdalas and P. Varbrand, A strongly polynomial algorithm for a concave production-transportation problem with a fixed number of nonlinear variables. Math. Programming 72 (1996) 229-258. | MR 1387761 | Zbl 0853.90116

[21] R. Wong, Introduction and recent advances in network design: Models and algorithms, in Transportation Planning Models, edited by M. Florian. Elsevier-North-Holland Publ. (1984). | Zbl 0594.90086