Condizioni di convessità nella programmazione dinamica

Marro, Giovanni ; Rossi, Remo

Atti della Accademia nazionale dei Lincei. Rendiconti della Classe di scienze fisiche, matematiche e naturali, Série 8, Tome 54 (1973) no. 4, pp. 604-614.

Résumé

The convexity of return functions in dynamic programming implies the possibility of employing standard procedures of convex programming for the searches of minima of functions which are performed at every stage of the computational procedure. In the present work by means of a geometric approach are derived necessary and sufficient conditions for the convexity of return functions in dynamic optimization problems with bounded states and controls and in presence of isoperimetric constraints.

Bibliographie

[1] H. W. Kuhn e A. W. Tucker, Nonlinear programming, «Proceedings of the 2-nd Berkeley Symposium on Mathematical Statistics and Probability», University of California Press,, Berkeley, 481-492 (1951). | MR

[2] P. Wolfe, Computational techniques for nonlinear programs, «Princeton University Conference on Linear Programming» (1957).

[3] P. Wolfe, Recent development in non-linear programming, The RAND Corporation Santa Monica, P-2063 (1960).

[4] J. B. Rosen, Optimal control and convex programming, «Proceedings of the IBM Scientific Computing Symposium on Control Theory and Applications», Yorktown Heights, N.Y., 223-237 (1964). | MR

[5] G. B. Dantzig, Linear control processes and mathematical programming, «Journal SIAM on Control», 4 (1) (1966). | MR

[6] H. C. Torng, Optimization of discrete control systems through linear programming, «Journal of the Franklin Institute», 278 (1), 28-44 (1964).

[7] E. B. Lee, A sufficient condition in the theory of optimal control, «Journal SIAM on Control», I (3), 241-245 (1963). | MR | Zbl

[8] E. B. Lee, Linear optimal control problems with isoperimetric constraints, «IEEE Transactions», AC-12 (1), 87-90 (1967). | DOI | MR

[9] Y. C. Ho, A successive approximation technique for optimal control systems subject to input saturation, «Transactions of the ASME, Journal of Basic Engineering», 84 (1), 33-40 (1962). | MR

[10] J. B. Rosen, Iterative solution of nonlinear optimal control problems, «Journal SIAM on Control», 4 (1), 223-244 (1966). | MR | Zbl

[11] E. G. Gilbert, An iterative procedure for computing the minimum of a quadratic form on a convex set, «Journal SIAM on Control», 4 (1), 61-80 (1966). | MR | Zbl

[12] R. O. Barr, Computation of optimal controls on convex reachable sets, «Mathematical Theory of Control», Academic Press, New York, 63-70 (1967). | MR

[13] R. O. Barr e E. G. Gilbert, Some iterative procedures for computing optimal controls, «Proceedings of the 3-rd IFAC Symposium», London 1966. | MR

[14] R. O. Barr e E. G. Gilbert, Some efficient algorithms for a class of abstract optimization problems arising in optimal control, «IEEE Transactions», AC-14 (6), 640-652 (1969). | DOI | MR

[15] G. Marro e R. Rossi, Sulla ottimizzazione dei sistemi discreti. Parte I: Sistemi lineari con vincoli convessi, «Calcolo», 4 (3) (1967). | DOI | MR | Zbl

[16] G. Marro e R. Rossi, Sulla ottimizzazione dei sistemi discreti. Parte II: Sistemi non lineari, «Calcolo», 4 (4) (1967). | DOI | MR

[17] B. Bernholtz e L. J. Graham, Hydrothermal economic scheduling. Part I: Solution by incremental dynamic programming, «AIEE Transactions», part. III, 79, 921-931 (1960).

[18] B. Bernholtz e L. J. Graham, Hydrothermal economic scheduling. Part II: Extension of the basic theory, «AIEE Transactions», part III, 80, 1089-1096 (1962).

[19] B. Bernholtz e L. J. Graham, Hydrothermal economic scheduling. Part III: Scheduling the thermal subsystem using constrained steepest descent, «AIEE Transactions », part III, 80, 1096-1105 (1962).

[20] B. Bernholtz e L. J. Graham, Hydrothermal economic scheduling. Part IV: A continuous procedure for maximizing the weighted output of a hydrolectric generating station, «AIEE Transactions», part III, 80, 1105-1107 (1962).