Partial Derivatives and Optimization Problem on Time Scales

The optimization problem using time scales is studied. Time scale is a model of time. The language of time scales seems to be an ideal tool to unify the continuous-time and the discrete-time theories. In this work we present necessary conditions for a solution of an optimization problem on time scales. To obtain that result we use properties and results of the partial diamond-alpha derivatives for continuous-multivariable functions. These results are also presented here.

Authors:

• Francisco Miranda

Keywords:

• Lagrange multipliers
• mathematical programming
• optimization problem
• time scales.

References:

[1] B. Aulbach and S. Hilger, "A unified approach to continuous and discrete
dynamics," in Qualitative theory of differential equations (Szeged, 1988),
vol. 53 of Colloq. Math. Soc. J'anos Bolyai, North-Holland, Amsterdam,
1990, pp. 37-56.
[2] S. Hilger, "Ein maßkettenkalk┬¿ul mit anwendung auf zentrumsmannigfaltigkeiten,"
Ph.D. dissertation, Universit¨at W¨urzburg, Germany, 1988.
[3] B. Jackson, "Partial dynamic equations on time scales," J. Comput. Appl.
Math., vol. 186, no. 2, pp. 391-415, 2006.
[4] M. Bohner and G. SH. Guseinov, "Partial differentiation on time scales,"
Dynam. Systems Appl., vol. 13, pp. 351-379, 2004.
[5] M. Z. Sarikaya, N. Aktan, H. Yildirim, and K. İIlarslan, "Partial Δ-
differentiation for multivariable functions on n-dimensional time scales,"
J. Math. Inequal., vol. 3, no. 2, pp. 277-291, 2009.
[6] P. Stehl'─▒k, "Maximum principles for elliptic dynamic equations," Math.
Comput. Model., vol. 51, no. 9-10, pp. 1193-1201, 2010.
[7] U. M. O¨ zkan and B. Kaymakalan, "Basics of diamond-α partial dynamic
calculus on time scales," Math. Comput. Model., vol. 50, no. 9-10, pp.
1253-1261, 2009.
[8] A. B. Malinowska and D. F. M. Torres, "Strong minimizers of the calculus
of variations on time scales and the Weierstrass condition," in Proc.
Estonian Acad. Sci. Phys. Math., vol. 58, no. 4, 2009, pp. 205-212.
[9] A. B. Malinowska and D. F. M. Torres, "The delta-nabla calculus of
variations," Fasc. Math., vol. 44, pp. 75-83, 2010.
[10] A. B. Malinowska and D. F. M. Torres, "Leitmann-s direct method of
optimization for absolute extrema of certain problems of the calculus
of variations on time scales," Appl. Math. Comput., vol. 217, no. 3, pp.
1158-1162, 2010.
[11] M. Bohner, "Calculus of variations on time scales," Dynam. Systems
Appl., vol. 13, no. 3-4, pp. 339-349, 2004.
[12] N. Martins and D. F. M. Torres, "Calculus of variations on time scales
with nabla derivatives," Nonlinear Anal., vol. 71, no. 12, pp. e763-e773,
2009.
[13] R. A. C. Ferreira, A. B. Malinowska, and D. F. M. Torres, "Optimality
conditions for the calculus of variations with higher-order delta derivatives,"
Appl. Math. Lett., vol. 24, no. 1, pp. 87-92, 2011.
[14] R. Hilscher, "Optimality conditions for time scale variational problems,"
D.Sc. dissertation, Masaryk University, Brno, Duben, 2008.
[15] M. Adivar and S.-C. Fang, "Convex optimization on mixed domains,"
J. Ind. Manag. Optim., vol. 8, no. 1, pp. 189-227, 2012.
[16] M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An
Introduction with Applications. Boston, MA: Birkh¨auser Boston, Inc.,
2001.
[17] M. Bohner and A. Peterson (Eds.), Advances in Dynamic Equations on
Time Scales. Boston, MA: Birkh¨auser Boston, Inc., 2003.
[18] J. W. Rogers Jr. and Q. Sheng, "Notes on the diamond-╬▒ dynamic
derivative on time scales," J. Math. Anal., vol. 326, no. 1, pp. 228-241,
2007.
[19] R. K. Sundaram, A First Course in Optimization Theory. Cambridge,
UK: Cambridge University Press, 1996.
[20] A. B. Malinowska and D. F. M. Torres, "The diamond-alpha Riemann
integral and mean value theorems on time scales," Dynam. Systems Appl.,
vol. 18, pp. 469-482, 2009.