Abstract
The recent algorithm proposed by the author for convex nonsmooth optimization is tailored to the case where the objective function is additive. We highlight aspects where this specialization differs from ignoring the structure of the objective function. To assess the approach, we consider two different nonlinear multicommodity flows applications in telecommunications including some comparison with a proximal bundle algorithm.
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Notes
The implementation of bfull is exactly the same as in [27] except that we use here a different version of CPLEX which results in slight differences in term of oracle calls.
References
Babonneau, F., Vial, J.-P.: ACCPM with a nonlinear constraint and an active set strategy to solve nonlinear multicommodity flow problems. Math. Program. B 120(1), 179–210 (2009)
Bacaud, L., Lemaréchal, C., Renaud, A., Sagastizábal, C.: Bundle methods in stochastic optimal power management: a disaggregated approach using preconditioners. Comput. Optim. Appl. 20, 227–244 (2001)
Ben Ameur, W., Ouorou, A.: Mathematical models of the delay constrained routing problem. Algorithm. Oper. Res. 1(2), 94–103 (2006)
Bertsekas, D.P., Gallager, R.G.: Data Networks. Prentice-Hall, Englewood Cliffs (1987)
Bertsekas, D.P., Tseng, P.: Partial proximal minimization algorithms for convex programming. SIAM J. Optim. 4(3), 551–572 (1994)
Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)
Cheney, E., Goldstein, A.: Newton’s method for convex programming and Tchebycheff approximations. Numeriche Mathematik 1, 253–268 (1959)
Eckstein, J., Fukushima, M.: Some reformulations and applications of the alternating direction method of multipliers. In: Large Scale Optimization: State of the Art. Kluwer Academic Publishers B.V., The Netherlands, pp. 119–138 (1993)
Elzinga, J., Moore, T.G.: A central cutting plane algorithm for the convex programming problem. Math. Program. 8, 134–145 (1975)
Emiel, G., Sagastizábal, C.: Incremental-like bundle methods with application to energy planning. Comput. Optim. Appl. 46, 305–332 (2010)
Frangioni, A.: Solving semidefinite quadratic problems within nonsmooth optimization algorithms. Comput. OR 23(11), 1099–1118 (1996)
Frangioni, A.: Dual-ascent methods and multicommodity flow problems. Ph.D. Thesis, Università di Pisa, Italy (1997)
Frangioni, A.: Generalized bundle methods. SIAM J. Optim. 13(1), 117–156 (2002)
Frangioni, A., Gorgone, E.: Generalized bundle methods for sum-functions with “easy” components: applications to multicommodity network design. http://www.optimization-online.org/DB_HTML/2011/06/3049.html
Goffin, J.-L., Vial, J.-Ph.: Convex nondifferentiable optimization: a survey focused on the analytic center cutting plane method. Optim. Methods Softw. 17(5), 805–867 (2002)
Goffin, J.-L., Gondzio, J., Sarkissian, R., Vial, J.-Ph.: Solving nonlinear multicommodity flow problems by the analytic center cutting plane method. Mathe. Program. 76(1), 131–154 (1997)
Hijazi, H., Bonami, P., Cornuejols, G., Ouorou, A.: Mixed integer nonlinear programs featuring “on/off” constraints: convex analysis and applications. Comput. Optim. Appl. (to appear)
Hiriart-Urruty, J.-B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms. Springer, Berlin (1993)
Kelley, J.E.: The cutting plane method for solving convex programs. J. Soc. Ind. Appl. Math. 8, 703–712 (1960)
Kelly, F.: Charging and rate control for elastic traffic. Eur. Trans. Telecommun. 8, 33–37 (1997)
Kiwiel, K.C.: Methods of descent for nondifferentiable optimization. Lecture Notes in Mathematics. Springer (1985)
Kiwiel, K.C.: Decomposition method of descent for minimizing the sum of convex nonsmooth functions. J. Optim. Theory Appl. 52(2), 255–271 (1987)
Kiwiel, K.C.: Proximity control in bundle methods for convex nondifferentiable minimization. Math. Program. 46(1), 105–122 (1990)
Kiwiel, K.C.: A Cholesky dual method for proximal piecewise linear programming. Numerische Mathematik 68(3), 325–340 (1994)
Kiwiel, K.C.: An alternating linearization bundle method for convex optimization and nonlinear multicommodity flow problems. Math. Program. 130, 59–84 (2011)
Lemaréchal, C., Sagastizábal, C.: Variable metric bundle methods: from conceptual to implementable forms. Math. Program. 76, 393–410 (1997)
Lemaréchal, C., Ouorou, A., Petrou, G.: A bundle-type algorithm for routing in telecommunication data networks. Comput. Optim. Appl. 44(3), 385–409 (2009)
Mahey, P., Ouorou, A., LeBlanc, L.J., Chifflet, J.: A new proximal decomposition algorithm for routing in telecommunication networks. Networks 31, 227–238 (1998)
Nedić, A., Bertsekas, D.: Incremental subgradient methods for nondifferentiable optimization. SIAM J. Optim. 12, 109–138 (2002)
Ouorou, A., Mahey, P., Vial, J.-Ph.: A survey of algorithms for convex multicommodity flow problems. Manage. Sci. 46(1), 126–147 (2000)
Ouorou, A.: A Proximal cutting plane method using Chebychev center for nonsmooth convex optimization. Math. Program. 119, 239–271 (2009)
Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)
Schramm, H., Zowe, J.: A version of the bundle idea for minimizing a nonsmooth function: conceptual ideas, convergence analysis, numerical results. SIAM J. Optim. 2(1), 121–152 (1992)
Acknowledgments
I’m indebted to the reviewers and the associate editors for their comments and suggestions which greatly improve an earlier version of this paper. I’m grateful to Antonio Frangioni for interesting and useful discussions on aggregation strategies.
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Dedicated to Claude Lemaréchal on the occasion of his 65th birthday.
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Ouorou, A. The proximal Chebychev center cutting plane algorithm for convex additive functions. Math. Program. 140, 163–187 (2013). https://doi.org/10.1007/s10107-012-0630-z
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DOI: https://doi.org/10.1007/s10107-012-0630-z
Keywords
- Nondifferentiable optimization
- Network flows
- Large-scale convex programming
- Routing problems in telecommunications