Sufficient Fritz John Type Optimality Criteria and Duality for Control Problems

Iqbal Husain, Santosh K. Srivastav

.
Department of Mathematics, Jaypee University of Engineering and Technology, Guna, India.

DOI: 10.4236/ajibm.2013.32029                

Abstract

Sufficient Fritz John optimality conditions are obtained for a control problem in which objective functional is pseudoconvex and constraint functions are quasiconvex or semi-strictly quasiconvex. A dual to the control problem is formulated using Fritz John type optimality criteria instead of Karush-Kuhn-Tucker optimality criteria and hence does not require a regularity condition. Various duality results amongst the control problem and its proposed dual are validated under suitable generalized convexity requirements. The relationship of our duality results to those of a nonlinear programming problem is also briefly outlined.

Keywords

Sufficient Optimality Criteria; Control Problem; Fritz John Type Dual; Nonlinear Programming Problem; Generalized Convexity

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	L. D. Berkovitz, “Variational Methods in Problems of Control and Programming,” Journal of Mathematical Analysis and Applications, Vol. 3, No. 1, 1961, pp. 145-169. doi:10.1016/0022-247X(61)90013-0
[2]	B. Mond and M. A. Hanson, “Duality for Control Problems,” SIAM Journal on Control, Vol. 6, 1965, pp. 28-35.
[3]	S. Chandra, B. D. Craven and I. Husain “A Class of Nondifferentiable Control Problems,” Journal of Optimization Theory and Applications, Vol. 56, No. 2, 1988, pp. 227-243. doi:10.1007/BF00939409
[4]	B. D. Craven, “Mathematical Programming and Control Theory,” Chapman and Hall, London, 1978. doi:10.1007/978-94-009-5796-1
[5]	F. A. Valentine, “The Problem of Lagrange with Differential Inequalities as Added Side Conditions, Contribution to the Calculus of Variation, 1933-1937,” University of Chicago Press, Chicago, 1937, pp. 407-448.
[6]	B. Mond and I. Smart, “Duality and Sufficiency in Control Problems with Invexity,” Journal of Mathematical Analysis and Applications, Vol. 136, No. 15, 1988, pp. 325-333.
[7]	I. Husain, A. Ahmed and B. Ahmad, “Sufficiency and Duality in Control Problems with Generalized Invexity,” Journal of Applied Analysis, Vol. 14 No. 1, 2008, pp. 27-42. doi:10.1515/JAA.2008.27
[8]	B. Mond and T. Weir, “Generalized Concavity and Duality in Generalized Concavity in Optimization and Economics,” In: S. Schaibl and W. T. Ziemba, Eds., Academic Press, New York, 1981, pp. 263-279.
[9]	I. Husain and S. K. Shrivastav, “Fritz John Duality in the Presence of Equality and Inequality Constarints,” Applied Mathematics, Vol. 3, No. 9, 2012, pp. 1023-1028. doi:10.4236/am.2012.39151
[10]	T. Weir and B. Mond, “Sufficient Fritz John Optimality Conditions and Duality for Nonlinear Programming Problems,” Opsearch, Vol. 23, No. 3, 1986, pp. 129-141.