On a Control Problem Containing Support Functions
Affiliation(s)
1 Department of Mathematics, Jaypee University of Engineering and Technology, Guna, India. 2 Department of Statistics, University of Kashmir, Srinagar, India.
1 Department of Mathematics, Jaypee University of Engineering and Technology, Guna, India. 2 Department of Statistics, University of Kashmir, Srinagar, India.
ABSTRACT
A control problem containing support functions in the integrand of the objective of the functional as well as in the inequality constraint function is considered. For this problem, Fritz John and Karush-Kuhn-Tucker type necessary optimality conditions are derived. Using Karush-Kuhn-Tucker type optimality conditions, Wolfe type dual is formulated and usual duality theorems are established under generalized convexity conditions. Special cases are generated. It is also shown that our duality results have linkage with those of nonlinear programming problems involving support functions.
A control problem containing support functions in the integrand of the objective of the functional as well as in the inequality constraint function is considered. For this problem, Fritz John and Karush-Kuhn-Tucker type necessary optimality conditions are derived. Using Karush-Kuhn-Tucker type optimality conditions, Wolfe type dual is formulated and usual duality theorems are established under generalized convexity conditions. Special cases are generated. It is also shown that our duality results have linkage with those of nonlinear programming problems involving support functions.
KEYWORDS
Control Problem, Support Function, Optimality Conditions, Generalized Convexity, Wolfe Type Duality, Nonlinear Programming Problem
Control Problem, Support Function, Optimality Conditions, Generalized Convexity, Wolfe Type Duality, Nonlinear Programming Problem
Cite this paper
Husain, I. , Ahmed, A. and Shah, A. (2014) On a Control Problem Containing Support Functions. American Journal of Operations Research, 4, 319-330. doi: 10.4236/ajor.2014.45031.
Husain, I. , Ahmed, A. and Shah, A. (2014) On a Control Problem Containing Support Functions. American Journal of Operations Research, 4, 319-330. doi: 10.4236/ajor.2014.45031.
References
[1] Mond, B. and Hanson, M.A. (1968) Duality for Control Problems. SIAM Journal on Control and Optimization, 6, 114-120. [2] Chandra, S., Craven, B.D. and Husain, I. (1988) A Class of Nondifferentiable Control Problem. Journal of Optimization Theory and ApplicationsJournal of Optimization Theory and Applications, 56, 227-243.
http://dx.doi.org/10.1007/BF00939409 [3] Hanson, M.A. (1968) Bounds for Functionally Convex Optimal Control Problems. Journal of Mathematical Analysis and Applications, 8, 84-89.
http://dx.doi.org/10.1016/0022-247X(64)90086-1 [4] Pearson, J.D. (1965) Reciprocality and Duality in Control Problems. Journal of Mathematical Analysis and Applications, 10, 383-408.
http://dx.doi.org/10.1016/0022-247X(65)90134-4 [5] Ringlee, R.J. (1965) Bounds for Convex Variational Programming Problems Arising in Power System, Scheduling and Control. IEEE Transaction on Automatic Control, 10, 28-35.
http://dx.doi.org/10.1109/TAC.1965.1098077 [6] Clarke, F.H. (1983) Optimization and Nonsmooth Analysis. Wiley, New York. [7] Demyanov, V.F. and Dixon, L.C.W. (1986) Quasidifferential Calculus. North-Holland, Amsterdam.
http://dx.doi.org/10.1007/BFb0121132 [8] Mond, B. and Schechter, M. (1996) Nondifferentiable Symmetric Duality. Bulletin of the Australian Mathematical Society, 53, 177-188.
http://dx.doi.org/10.1017/S0004972700016890 [9] Craven, B.D. and Mond, B. (1979) Langrangean Conditions for Quasi-Differentiable Optimization. Proceedings of the 9th International Conference on Mathematical Programming, Akademiai Kiado, Budapest and North-Holand, Amstedam. [10] Kreindler, F. (1966) Reciprocal Optimal Control Problems. Journal of Mathematical Analysis and Applications, 14, 141-152.
http://dx.doi.org/10.1016/0022-247X(66)90067-9 [11] Husain, I. and Jabeen, Z. (2008) Continuous Programming Containing Support Functions. Journal of Applied Mathematics and Informatics, 26, 75-106. [12] Husain, I., Abha and Jabeen, Z. (2002) On Nonlinear Programming with Support Functions. Journal of Applied Mathematics and Computing, 10, 83-99.
[1] Mond, B. and Hanson, M.A. (1968) Duality for Control Problems. SIAM Journal on Control and Optimization, 6, 114-120. [2] Chandra, S., Craven, B.D. and Husain, I. (1988) A Class of Nondifferentiable Control Problem. Journal of Optimization Theory and ApplicationsJournal of Optimization Theory and Applications, 56, 227-243.
http://dx.doi.org/10.1007/BF00939409 [3] Hanson, M.A. (1968) Bounds for Functionally Convex Optimal Control Problems. Journal of Mathematical Analysis and Applications, 8, 84-89.
http://dx.doi.org/10.1016/0022-247X(64)90086-1 [4] Pearson, J.D. (1965) Reciprocality and Duality in Control Problems. Journal of Mathematical Analysis and Applications, 10, 383-408.
http://dx.doi.org/10.1016/0022-247X(65)90134-4 [5] Ringlee, R.J. (1965) Bounds for Convex Variational Programming Problems Arising in Power System, Scheduling and Control. IEEE Transaction on Automatic Control, 10, 28-35.
http://dx.doi.org/10.1109/TAC.1965.1098077 [6] Clarke, F.H. (1983) Optimization and Nonsmooth Analysis. Wiley, New York. [7] Demyanov, V.F. and Dixon, L.C.W. (1986) Quasidifferential Calculus. North-Holland, Amsterdam.
http://dx.doi.org/10.1007/BFb0121132 [8] Mond, B. and Schechter, M. (1996) Nondifferentiable Symmetric Duality. Bulletin of the Australian Mathematical Society, 53, 177-188.
http://dx.doi.org/10.1017/S0004972700016890 [9] Craven, B.D. and Mond, B. (1979) Langrangean Conditions for Quasi-Differentiable Optimization. Proceedings of the 9th International Conference on Mathematical Programming, Akademiai Kiado, Budapest and North-Holand, Amstedam. [10] Kreindler, F. (1966) Reciprocal Optimal Control Problems. Journal of Mathematical Analysis and Applications, 14, 141-152.
http://dx.doi.org/10.1016/0022-247X(66)90067-9 [11] Husain, I. and Jabeen, Z. (2008) Continuous Programming Containing Support Functions. Journal of Applied Mathematics and Informatics, 26, 75-106. [12] Husain, I., Abha and Jabeen, Z. (2002) On Nonlinear Programming with Support Functions. Journal of Applied Mathematics and Computing, 10, 83-99.