Fritz John Duality in the Presence of Equality and Inequality Constraints

Affiliation(s)

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

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

ABSTRACT

A dual for a nonlinear programming problem in the presence of equality and inequality constraints which represent many realistic situation, is formulated which uses Fritz John optimality conditions instead of the Karush-Kuhn-Tucker optimality conditions and does not require a constraint qualification. Various duality results, namely, weak, strong, strict-converse and converse duality theorems are established under suitable generalized convexity. A generalized Fritz John type dual to the problem is also formulated and usual duality results are proved. In essence, the duality results do not require any regularity condition if the formulations of dual problems uses Fritz John optimality conditions.

A dual for a nonlinear programming problem in the presence of equality and inequality constraints which represent many realistic situation, is formulated which uses Fritz John optimality conditions instead of the Karush-Kuhn-Tucker optimality conditions and does not require a constraint qualification. Various duality results, namely, weak, strong, strict-converse and converse duality theorems are established under suitable generalized convexity. A generalized Fritz John type dual to the problem is also formulated and usual duality results are proved. In essence, the duality results do not require any regularity condition if the formulations of dual problems uses Fritz John optimality conditions.

Cite this paper

I. Husain and S. Shrivastav, "Fritz John Duality in the Presence of Equality and Inequality Constraints,"*Applied Mathematics*, Vol. 3 No. 9, 2012, pp. 1023-1028. doi: 10.4236/am.2012.39151.

I. Husain and S. Shrivastav, "Fritz John Duality in the Presence of Equality and Inequality Constraints,"

References

[1] R. W. Cottle, “A Theorem of Fritz John in Mathematical Programming,” RAND Memorandum RM-3538-PR, 1963.

[2] F. John, “Extremum Problems with Inequalities as Side Condition,” In: K. O. Frierichs, O. E. Neugebaur and J. J. Stoker, Eds., Studies and Essays, Courant Anniversary Volume, Wiley (Interscience), New York, 1984, pp. 187204.

[3] 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.

[4] O. L. Mangasarian and S. Fromovitz, “The Fritz John Necessary Optimality Conditions in the Presence of Equality and Inequality Constraints,” Journal of Mathematical Analysis and Applications, Vol. 17, No. 1, 1967, pp. 37-47. doi:10.1016/0022-247X(67)90163-1

[5] H. W. Kuhn and A. W. Tucker, “Nonlinear Programming,” Proceeding of the Second Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Berkeley, 1951, pp. 481-492.

[6] O. L. Mangasarian, “Nonlinear Programming,” McGrawHill, New York, 1969.

[1] R. W. Cottle, “A Theorem of Fritz John in Mathematical Programming,” RAND Memorandum RM-3538-PR, 1963.

[2] F. John, “Extremum Problems with Inequalities as Side Condition,” In: K. O. Frierichs, O. E. Neugebaur and J. J. Stoker, Eds., Studies and Essays, Courant Anniversary Volume, Wiley (Interscience), New York, 1984, pp. 187204.

[3] 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.

[4] O. L. Mangasarian and S. Fromovitz, “The Fritz John Necessary Optimality Conditions in the Presence of Equality and Inequality Constraints,” Journal of Mathematical Analysis and Applications, Vol. 17, No. 1, 1967, pp. 37-47. doi:10.1016/0022-247X(67)90163-1

[5] H. W. Kuhn and A. W. Tucker, “Nonlinear Programming,” Proceeding of the Second Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Berkeley, 1951, pp. 481-492.

[6] O. L. Mangasarian, “Nonlinear Programming,” McGrawHill, New York, 1969.