I propose an alternative method for computing effectively the solution of the control inventory problem under non-convex polynomial cost functions. I apply the method of moments in global optimization to transform the corresponding, non-convex dynamic programming problem into an equivalent optimal control problem with linear and convex structure. I device computational tools based on convex optimization, to solve the convex formulation of the original problem.

Key words: Method of moments, control of inventories, non-convex polynomial functions, global optimization.

Propongo un método alternativo para calcular de manera efectiva la solución del problema de control de inventarios bajo funciones de costo polinomiales no convexas. Aplico el método de momentos en optimización global para transformar el correspondiente problema de programación dinámica no convexo en un problema de control óptimo equivalente con estructura lineal y convexa. Diseño herramientas computacionales basada en optimización convexa para resolver la formulación convexa del problema original.

Palabras clave: Método de momentos, control de inventorias, funciones polinomiales no convexas, optimización global.

Texto completo disponible en PDF

References

1 N. I. Akhiezer, Some questions in the theory of moments, Translation of Mathematical Monographs, Amer. Mathematical Society 1962.         [ Links ]

2 M.S. Bazaraa, H. D. Sherali & C. M. Shetty, Non linear programming, 2 ed. John Wiley & Sons, New York, 1993.         [ Links ]

3 A. Ben-Tal & A. Nemirovski, Lectures on modern convex optimization, TMPS, SIAM 2001.         [ Links ]

4 D. Bertsekas, Non linear programming note, Athenea Sci. 1999.         [ Links ]

5 D. Bertsekas, Dynammic programming and optimal control, Vol I, 2nd edition, Massachusets Institute Technology, Athenea Scientific, Belmont, Massachusetts 2001.         [ Links ]

6 A. Blinder, Can the production smoothing model of inventory behavior be saved?, The Quartery Journal of Economics, 101 (1986) 3, 431-454.         [ Links ]

7 T. Bresnahan & V. Ramey, Output fluctuations at the plant level, The Quarterly Journal of Economics, 109 (1986) 3, 593-624.         [ Links ]

8 E. Cerda, Optimización dinámica, Universidad Complutense de Madrid, Prentice Hall, Spain, 2001.         [ Links ]

9 A. Conn, N. Gould & P. Toint, Trust-region methods, SIAM 2000.         [ Links ]

10 R. Cooper & J. Haltinwager, The aggregate implications of machine replacement: theory and evidence, The American Economic Review, 83 (1993) 3, 360-382.         [ Links ]

11 R. Curto & L. Fialkow, Recursiveness, positivity, and truncated moment problems, Houston Journal of Mathematics, 17(1991) 4, 603-635.         [ Links ]

12 D. Escobar, Introducción a la economía matemática, Grupo Editorial Iberoamérica. Bogotá, Colombia 1998.         [ Links ]

13 M. Fafchamps, H. Gunning Han & R. Oostendorp, Inventories and risk in african manufacturing, Economic Journal, 110 (1999), 861-893.         [ Links ]

14 R. Flectcher, Practical methods of optimization, John Wiley & Sons 1987.         [ Links ]

15 G. Hall, Non-convex costs and capital utilization: a study of production scheduling at automobile asembly plants, Journal of Monetary Economics, 45 (2000), 681-716.         [ Links ]

16 M. G. Krein & A. A. Nudelman, The Markov moment problem and extremal problems, Translations of Mathematical Monographs, 50 1977, Amer. Mathematical Society.         [ Links ]

17 L. Laserre, Global optimization with polynomial and the problem of moments, SIAM J. Optim. 11 (2001) 3, 796-817.         [ Links ]

18 J. Laserre, Semidefinite programming vs lp relaxation for polynomial programming, Annals of Operation Research Journal, 27 (2002) 347-360.         [ Links ]

19 D. Luenberger, Programación lineal y no lineal, Addison-Wesley, Buenos Aires, 1989.         [ Links ]

20 O. L. Mangasarian, Nonlinear programming, SIAM, Philadelphia, 1994.         [ Links ]

21 MathWorks, The MathWorks Inc, Optimization Toolbox for Use With Matlab. User's Guide 2000.         [ Links ]

22 R. Meziat, P. Pedregal & J. Egozcue, The method of moments for non- convex variational problems, in Advances in convex analysis and global optimization, non convex optimization and its applications series, 54 (2001), 371-381.         [ Links ]

23 R. Meziat, The method of moments in global optimization, Journal of Mathematical Sciences, 116 (2003), 3033-3324.         [ Links ]

24 R. Meziat, Analysis of non convex polynomial programs by the method of moments, Nonconvex optimization and its application series 74 (2003), 353-372.         [ Links ]

25 R. Meziat, Analysis of two dimensional non convex variational problems, Optimization and Control with Applications. Qi Teo and Yan eds. Published by Kluwer 2003.         [ Links ]

26 R. Meziat & J. Villalobos, Analysis of microestructures and phase transition phenomena in one-dimensional non-linear elasticity by convex optimization, in press. Variational Methods and Optimization 3 (2005), Omeva.         [ Links ]

27 R. Meziat, D. Patiño & P. Pedregal, An alternative approach for non-linear optimal control problems based on the method of moments, in press.         [ Links ]

28 Y. Nesterov, Squared functional systems and optomization problems in high performance optimization, Frenk, H., K. Roos and T. Terlaky, eds., Kluwer.         [ Links ]

29 V. Ramey, Nonconvex costs and the behavior of inventories, The Journal of Political Economy, 99 (1991) 2, 306-334.         [ Links ]

30 V. Ramey & D. Vine, Tracking the source of the decline in gdp volatility: an analysis of the automobile industry, Working Paper 10384, National Bureau of Economic Research 2004.         [ Links ]

31 T. Rockafellar, Convex analysis, Princeton University Press, Princeton, New Jersey 1970.         [ Links ]

32 N. Shor, Nondifferentiable optimization and polynomial problems, Kluwer, Boston-Dordrect-London, 1998.         [ Links ]

33 H. R Varian, Intermediate microeconomics: a modern approach, 4th edition. W.W. Norton & Company, London 1996.         [ Links ]