Abstract
We address a class of particularly hard-to-solve combinatorial optimization problems, namely that of multicommodity network optimization when the link cost functions are discontinuous step increasing. Unlike usual approaches consisting in the development of relaxations for such problems (in an equivalent form of a large scale mixed integer linear programming problem) in order to derive lower bounds, our d.c.(difference of convex functions) approach deals with the original continuous version and provides upper bounds. More precisely we approximate step increasing functions as closely as desired by differences of polyhedral convex functions and then apply DCA (difference of convex function algorithm) to the resulting approximate polyhedral d.c. programs. Preliminary computational experiments are presented on a series of test problems with structures similar to those encountered in telecommunication networks. They show that the d.c. approach and DCA provide feasible multicommodity flows x * such that the relative differences between upper bounds (computed by DCA) and simple lower bounds r:=(f(x*)-LB)/{f(x*)} lies in the range [4.2 %, 16.5 %] with an average of 11.5 %, where f is the cost function of the problem and LB is a lower bound obtained by solving the linearized program (that is built from the original problem by replacing step increasing cost functions with simple affine minorizations). It seems that for the first time so good upper bounds have been obtained.
Similar content being viewed by others
References
K.T. Ahuja, T. Magnanti and J. Orlin. Networks Flows: Theory, Algorithms and Applications. Prentice Hall, 1993.
A.A. Assad. Multicommodity network flows-A survey. Networks 8: 37–91, 1978.
A. Balakrisnan and S.C. Graves. A composite algorithm for a concave-cost network flow problem. Networks 19: 175–202, 1989.
D.P. Bertsekas and R.G. Gallager. Data Netwoks. Prentice Hall, 1987.
F. Boyer. Conception et Routage des Réseaux de Télécommunications. Thèse de Doctorat de l'Université Blaise Pascal, Clermont-Ferrand, 1997.
V.J.M. Ferreira Filho and R.D. Galvao. A Survey of Computer Network Design Problems. Investigacion Operativa 4: 183–211, 1994.
L. Fratta, M. Gerla and L. Kleinrock. The flow deviation method: an approach to store-andforward communications network design. Networks 3: 97–133, 1973.
V. Gabrel and M. Minoux. Large scale LP relaxations for minimum cost multicommodity flow problems with step increasing cost functions and computational results. Technical Report, Laboratoire MASI, Univ. Paris 6, 1996.
V. Gabrel and M. Minoux. LP relaxations better than convexification for multicommodity network optimization problems with step increasing cost functions. Acta Mathematica Vietnamica 22: 128–145, 1997.
B. Gavish. Augmented Lagrangian based bounds for centralized network design. IEEE Transactions on Communications COM-33, 1247–1257, 1985.
B. Gavish and K. Altinkemer. Backbone network design tools with economic tradeoffs ORSA J. on Computing 2/3: 236–252.
B. Gavish and G.W. Graves. System for routing and capacity assignment in computer communication network. IEEE Transactions on Communications COM-37: 360–366, 1989.
M. Gerla and L. Kleinrock. On the tological design of distributed computer networks. IEEE Transactions on Communications COM-25: 48–60, 1977.
M. Gerla The Design of Store-and-Forward Networks for Computer Communications. PhD Thesis, UCLA, 1973.
M. Gerla and L. Keinrock. On the topological design of distributed computer networks. IEEE Transactions on Communications COM-25: 48–60, 1977.
M. Gerla, R. Monteiro and R. Pazos, Topology design and bandwith allocation in ATM nets. IEEE transactions on selected Area in Communications SAC-7, 1989.
M. Gondran and M. Minoux. Graphes et Algorithmes Eyrolles, Paris 2nd edition 1995.
J.B. Hiriart Urruty and C. Lemarechal Convex Analysis and Minimization Algorithms. Springer, Berlin, 1993.
P.J. Laurent. Approximation et Optimisation. Hermann, Paris, 1972.
R. Horst, P.M. Pardalos and V.T. Nguyen. Introduction to Global Optimization, Kluwer, Dordrecht, 1995.
R. Horst and V.T. Nguyen. D.C. Programming: Overview. Journal of Optimization Theory and Applications, 103: 1–43, 1999.
J.L. Kennington. A survey of linear cost multicommodity network flows. Operations Research 26: 209–236, 1978.
H.A. Le Thi. Contribution à l'optimisation non convexe et l'optimisation globale: Théorie, Algorithmes et Applications. Habilitation à Diriger des Recherches, Université de Rouen, 1997.
H.A. Le Thi and T. Pham Dinh. Solving a class of linearly constrained indefinite quadratic problems by D.c. algorithms. Journal of Global Optimization 11: 253–285, 1997.
H.A. Le Thi and T. Pham Dinh. D.c. models of real world nonconvex optimization problems. Technical Report, LMI, INSA-Rouen, 1998.
H.A. Le Thi and T. Pham Dinh. D.c. programming approach for large-scale molecular optimization via the general distance geometry problem, in Optimization in Computational Chemistry and Molecular Biology: Local and Global Approaches, C.A. Floudas and P.M. Pardalos (Eds.), pp. 301–339, Kluwer Academic Publishers, Dordrecht, 2000.
H.A. Le Thi and T. Pham Dinh, A continuous approach for large-scale linearly constrained quadratic zero-one programming. Optimization, 45(3) 1–28, 2001.
J. Mac Gregor Smith and P. Winter (eds). Topological NetWork Design. Annals of Operations Research 33, 1991.
P. Mahey and T. Pham Dinh. Proximal decomposition on the graph of a maximal monotone operator. SIAM Journal on Optimization, 5: 454–468, 1995.
T.Q. Nguyen. Une approche D.c. en Optimisation dans les Réseaux. Algorithmes, Codes et Simulations Numériques. PhD Thesis, Université de Rouen, 1999.
P. Mahey, A. Ourou, L. Leblanc and J. Chifflet. A new proximal decomposition algorithm for routing in telecommunications networks. Networks 31: 227–338, 1998.
P. Mahey and H.P.L. Luna. Bounds for Global Optimization of Capacity Expansion and Flow Assignment Problems. to appear in Operations Research Letters.
P. Mahey and H.P.L. Luna. Separable convexification techniques for capacity and flow assignment problems. To appear.
V.T. Nguyen. D.C. Programming. in Encyclopedia of Optimization Kluwer Academic Publishers, Dordrecht.
M. Minoux. Network synthesis and Optimum Network design problems: Models, Solution Methods and Applications. Networks 19: 313–360, 1989.
A. Ouorou. Décomposition proximale des problèmes de multiflots à critère convexe-Applications aux problèmes de routage dans les réseaux de communications. Thèse de Doctorat, Université de Clermont-Ferrand, 1995.
T. Pham Dinh and H.A. Le Thi Convex analysis approach to d.c. programming: Theory, Algorithms and Applications (dedicated to Professor Hoang Tuy on the occasion of his 70th birthday). Acta Mathematica Vietnamica. 22(1): 289–355.
T. Pham Dinh and H.A. Le Thi. D.c. optimization algorithms for trust region problem. SIAM J. Optimization, 8(2): 476–505.
R.T. Rockafellar. Convex Analysis. Princeton University, Princeton 1970.
R.T. Rockafellar. Monotone operators and the proximal point algorithm. SIAM J. Control and Optimization, 14(5): 877–898.
B. Sanso, F. Soumis and M. Gendreau. On the evaluation of telecommunication networks reliability using routing models. IEEE Transactions on Communications COM-3, 1494–1501.
J.F. Toland. On subdifferential calculus and duality in nonconvex optimization. Bull. Soc. Math. France Mémoire 60: 173–180.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Hoai An, L.T., Tao, P.D. D.C. programming approach for multicommodity network optimization problems with step increasing cost functions. Journal of Global Optimization 22, 205–232 (2002). https://doi.org/10.1023/A:1013867331662
Issue Date:
DOI: https://doi.org/10.1023/A:1013867331662