Un algoritmo cuasi-newton con aproximaciones consistentes para programacion semi-infinita
DOI:
https://doi.org/10.22199/S07160917.1997.0002.00002Abstract
En este artículo desarrollamos un algoritmo cuasi-Newton de puntos interiores para la resolución de problemas semi-infinitos no lineales. Usamos una estrategia de discretización basada en el concepto de aproximaciones consistentes, la cual genera una sucesión de problemas aproximados que convergen Epigráficamente al problema original. Además se construye una función de optimalidad que resulta natural para métodos de Newton y se demuestra que todo punto de acumulación de la sucesión de puntos estacionarios correspondientes a la sucesión de problemas aproximados es un punto estacionario del problema original.
El uso de esta técnica de aproximaciones consistentes en conjunción con una estrategia de discretización diagonalizante y de filtrado de restricciones es ilustrada con ejemplos numéricos.
References
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[4] C. Gonzaga and E. Polak, On constraint dropping scheme and optimality functions for a class of outer approximations algorithms. Siam J. Control and Optimizations, Vol 17, No 4 (1979).
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[6] S. A. Gustafson, A tree phase algorithm for Semi Jnfinite Programs. Semi Infinite Programming and applications. Lectures notes in Economics and Mathematical Systems 215. Springer Verlag (1983).
[7] L. He and E. Polak, Effective diagonalizat-ion strategies for the solution of a class of Optimal Design Problerns. IEEE Trans. on automatic control Vol 35, No 3 (1990) J. Herskovits An interior point technique for Nonlinear Optimization. INRIA, modulef proj B.P.l05, 78153 Le Chesnay Cedex (1993).
[8] R. Hettich,
a) A comparison of some numerical methods for Semi lnfinite Programming. Semi lnfinite Programming, proceeding of a workshop Springer Verlag (1979).
b) A review of nnmerical methods for Semi lnfinite Optimization. Semi lnfinite Programming and application proceeding of an international. Symposium. Springer Verlag (1983).
c) An implernentation of a discretization method for Semi lnfinite Programming. Mathematical Programming 34 (1986).
[9] R. Hettich and G. Gramlich, A note on a implementation of a method for quadratic Semi Infinite Programming. Mathematical programming. 46 (1990).
[10] R. Hettich and W.van Honsted, On quadratically convergent methods for Semi Infinite Programming. In [9b].
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[12] R. Hettich and K.O Kortanek, Semi Infinite Programming. Theory methods and applications. Siam Review, Vol 35, No3 (1993).
[13] L. He and Polak, Effective diagonalization strategies for the solution of a class of Optimal Design Problems. IEEE Trans. on automatic control Vol 35, No3 (1990).
[14] W. van Honstede, An approximation method for Semi Infinite Problems. In [9b].
[15] H. Hu, A one phase algorithm for Semi Infinite Linear Programming. Mathematical Programming 46 (1990).
[16] H.T. Jongen, F. Twilt and G.W Weber, Semi Infinite Optimization struture and stability of the feasible set. JOTA Vol 72, No 3 (1992).
[17] R.Klessig and E.Polak, An adaptative precision Gradient method for Optimal Control. Siam J. Control, Vol 11, No 1 (1973).
[18] D.Q Mayne andE. Polak, A Quadratic convergent algorithm for solving Infinite dimensional Inequalities. Appl. Matematics and Optimization ( 1982).
[19] M. Minoux, Mathematical Programrning. Theory and Algorithms. Ed. John Wiley and Sons (1986).
[20] E. Panier and A. Tits, A globally convergent algorithm wdh adaptively refined discretization for Semi lnfinite Optimization Problems arising in Engineering Design. System Research Center and Electrical Engineering department. Univers. of Maryland. (1988).
[21] E. Polak,
a) Computational methods in Optimization. Academic Press –New York (1971).
b) On the Mathematical foundations of Nondifferentiable Optimization in Engineering Design. Siam Review, Vol 29, No1 (1987).
c) On the use of consistent approximations in the solution of Semi Infinite Optimization and Optimal Control Problems. Mathematical Programming 62, No 2 (1993).
[22] E. Polak and L. He,
a) Rate preserving discretization strategies for Semi /nfinite Programming and Optimal Control. Siam J. Control and Optimization, Vol 30, No 3 (1992).
b) A unified Phase-I Phase-II method of feasible directions for Semi-lnfinite Optimization. M No UCB/ERL M89/7,University of California, Berkeley ( 1989).
[23] E. Polak, D.Q. Mayne, and J. E. Higgins, On the extension of Newton method to Semi Infinite Minimax problems. Siam J. Control and Optimization, Vol 30, No2 (1992).
[24] E. Polak and A. Tits, A recursive quadratic programming algorithm for Semi Infinite Optimization Problems. Appl. Math. Optimization (1982).
[25] R. Reemtsen, Discretization methods for the solution of Semi Infinite Programming Problems. JOTA Vol 71, No1 (1991).
[26] Y. Tanaka, M. Fukushima, T. Ibaraki, A comparative study of several Semi-Infinite nonlinear Programming Algorithms. European Journal of Operational Research 36 (1988).
[27] G.A. Watson, Numerical experiments with globally convergent methods for Semi Infinite Programming Problems. In [9b]
[28] G.A. Watson and I.D. Coope, A projected Lagrangian algorithm for Semi Infinite Programming. Mathematical Programming 32 (1985).
[2] J.C. Dunn, Diagonally modified conditional Gradient methods for input constrained Optimal Control Problems. Siam J. Control and Optimization, Vol 24, No 6 (1986).
[3] K. Glashoff and Sven Ake Gustafson, Linear Optimization and approximation. Springer Verlag (1983).
[4] C. Gonzaga and E. Polak, On constraint dropping scheme and optimality functions for a class of outer approximations algorithms. Siam J. Control and Optimizations, Vol 17, No 4 (1979).
[5] C. Gonzaga, E. Polak and Trahan, An improved algor'ithm for Optimization Problems with functional lnequality constraints. IEEE Trans. Automat. control AC-25 (1980).
[6] S. A. Gustafson, A tree phase algorithm for Semi Jnfinite Programs. Semi Infinite Programming and applications. Lectures notes in Economics and Mathematical Systems 215. Springer Verlag (1983).
[7] L. He and E. Polak, Effective diagonalizat-ion strategies for the solution of a class of Optimal Design Problerns. IEEE Trans. on automatic control Vol 35, No 3 (1990) J. Herskovits An interior point technique for Nonlinear Optimization. INRIA, modulef proj B.P.l05, 78153 Le Chesnay Cedex (1993).
[8] R. Hettich,
a) A comparison of some numerical methods for Semi lnfinite Programming. Semi lnfinite Programming, proceeding of a workshop Springer Verlag (1979).
b) A review of nnmerical methods for Semi lnfinite Optimization. Semi lnfinite Programming and application proceeding of an international. Symposium. Springer Verlag (1983).
c) An implernentation of a discretization method for Semi lnfinite Programming. Mathematical Programming 34 (1986).
[9] R. Hettich and G. Gramlich, A note on a implementation of a method for quadratic Semi Infinite Programming. Mathematical programming. 46 (1990).
[10] R. Hettich and W.van Honsted, On quadratically convergent methods for Semi Infinite Programming. In [9b].
[11] R. Hettich and Jongen, Semi Infinite Programming, condition of optimality and applications. Optimization techniques, part 2. Lectures notes in control and information sciences. Springer Verlag 7 (1978).
[12] R. Hettich and K.O Kortanek, Semi Infinite Programming. Theory methods and applications. Siam Review, Vol 35, No3 (1993).
[13] L. He and Polak, Effective diagonalization strategies for the solution of a class of Optimal Design Problems. IEEE Trans. on automatic control Vol 35, No3 (1990).
[14] W. van Honstede, An approximation method for Semi Infinite Problems. In [9b].
[15] H. Hu, A one phase algorithm for Semi Infinite Linear Programming. Mathematical Programming 46 (1990).
[16] H.T. Jongen, F. Twilt and G.W Weber, Semi Infinite Optimization struture and stability of the feasible set. JOTA Vol 72, No 3 (1992).
[17] R.Klessig and E.Polak, An adaptative precision Gradient method for Optimal Control. Siam J. Control, Vol 11, No 1 (1973).
[18] D.Q Mayne andE. Polak, A Quadratic convergent algorithm for solving Infinite dimensional Inequalities. Appl. Matematics and Optimization ( 1982).
[19] M. Minoux, Mathematical Programrning. Theory and Algorithms. Ed. John Wiley and Sons (1986).
[20] E. Panier and A. Tits, A globally convergent algorithm wdh adaptively refined discretization for Semi lnfinite Optimization Problems arising in Engineering Design. System Research Center and Electrical Engineering department. Univers. of Maryland. (1988).
[21] E. Polak,
a) Computational methods in Optimization. Academic Press –New York (1971).
b) On the Mathematical foundations of Nondifferentiable Optimization in Engineering Design. Siam Review, Vol 29, No1 (1987).
c) On the use of consistent approximations in the solution of Semi Infinite Optimization and Optimal Control Problems. Mathematical Programming 62, No 2 (1993).
[22] E. Polak and L. He,
a) Rate preserving discretization strategies for Semi /nfinite Programming and Optimal Control. Siam J. Control and Optimization, Vol 30, No 3 (1992).
b) A unified Phase-I Phase-II method of feasible directions for Semi-lnfinite Optimization. M No UCB/ERL M89/7,University of California, Berkeley ( 1989).
[23] E. Polak, D.Q. Mayne, and J. E. Higgins, On the extension of Newton method to Semi Infinite Minimax problems. Siam J. Control and Optimization, Vol 30, No2 (1992).
[24] E. Polak and A. Tits, A recursive quadratic programming algorithm for Semi Infinite Optimization Problems. Appl. Math. Optimization (1982).
[25] R. Reemtsen, Discretization methods for the solution of Semi Infinite Programming Problems. JOTA Vol 71, No1 (1991).
[26] Y. Tanaka, M. Fukushima, T. Ibaraki, A comparative study of several Semi-Infinite nonlinear Programming Algorithms. European Journal of Operational Research 36 (1988).
[27] G.A. Watson, Numerical experiments with globally convergent methods for Semi Infinite Programming Problems. In [9b]
[28] G.A. Watson and I.D. Coope, A projected Lagrangian algorithm for Semi Infinite Programming. Mathematical Programming 32 (1985).
Published
2018-04-04
How to Cite
[1]
C. N. Baracatt and J. Heskovits N., “Un algoritmo cuasi-newton con aproximaciones consistentes para programacion semi-infinita”, Proyecciones (Antofagasta, On line), vol. 16, no. 2, pp. 99-124, Apr. 2018.
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