Fritz John Type Duality in Nondifferentiable Continuous Programming with Equalit

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Abstract： A fritz john type dual for a nondifferentiable continuous programming problem with equality and inequality constraints which represent many realistic situations is formulated using Fritz John type optimality conditions instead of Karush-Kuhn-Tucker type conditions and thus does not require a regularity condition. Various duality results under suitable generalized convexity assumptions are derived. A pair of Fritz John type dual continuous programming with natural boundary conditions rather thanfixed end points is also presented. Finally， it is pointed that our duality results can be considered as dynamic generalizations of those of a nondifferentiable nonlinear programming problem in the presence of equality and inequality constraints recently treated in the literature.

Key words： Fritz John type duality； Semi-strict pseudoconvexity； Nonlinear programming； Natural boundary conditions

1. INTRODUCTION

Optimality conditions form the foundations of mathematical programming both theoretically and computationally. The best known necessarily optimality condition for a mathematical programming problem is the Karush-Kuhn-Tucker optimality condition. However， the Fritz John optimality conditions which appeared before the Karush-Kuhn-Tucker optimality conditions by about three years， is regarded more general in a sense. In order for Karush-Kuhn-Tucker optimality criterion to hold， a constraint-qualification on the constraint functions of the problem is required to be imposed while no such qualification need be imposed on the constraints for the Fritz John optimality conditions to hold. In literature of mathematical programming， Karush-Kuhn-Tucker optimality conditions are generally used to formulate for Wolfe and Mond-Weir type duals， thus a constraint qualification is needed in order to eliminate the requirement of constraint qualification， Weir and Mond [1] used Fritz John optimality condition instead of Karush-Kuhn-Tucker optimality conditions to study duality for a constrained nonlinear programming optimization problem.

Hanson [2] pointed out that some of the duality theorems of mathematical programming have analogues in the variational calculus. This relationship between mathematical programming and the classical calculus of variations is explored and extended. Since mathematical programming and the classical calculus of variations have undergone independent development， it is felt that the mutual adaption of ideas and techniques may prove useful. Mond and Hanson [3] were thefirst two formulate a dual problem for constrained variational problem and established duality results using a regularity condition. Subsequently Chandra et al. [4] studied Wolfe type duality for a class of nondifferentiable continuous programming problems while Bector et al. [5] studied Mond-Weir type duality for the problem of [4]. In order to derive duality results for the variational problem， they required a constraint qualification.

Motivated with the results of Weir and Mond [1]， in this paper we study Fritz john type duality for a class of nondifferentiable continuous programming problems with equalities and inequalities together which represent many realistic situations. A pair of Fritz John type dual continuous programming with natural boundary conditions rather thanfixed end points is also presented. The relationship between our results and those of [1] is also indicated.

2. STATEMENT OF THE PROBLEMS AND PRELIMINARIES

Consider the following non-differentiable continuous programming problems treated in [4]：（CEP）： Minimize

with equality if B （t）[x（t）？q （t） z （t）] = 0 for some q（t）∈R.

3. FRITZ JOHN TYPE DUALITY

Using Karush-Kuhn-Tuck optimality conditions， Chandra et al. [4] have formulated the following Wolfe type dual continuous programming problem for the problem（CP）：

（WCED）： Maximize

Strong duality for both the Wolfe type dual and MondWeir type dual is established using Karush-Kuhn-Tucker necessary optimality conditions at the optimum of （CP）.

Here we consider （CEP） and propose a different dual to establish strong duality using Fritz John type necessary optimality conditions. Consequently no regularity condition is required.

The following is the Fritz John type dual to the problem （CEP）：

（FrCED）： Maximize

Contradicting the feasibility of （r，ˉx，ˉy，ˉz，ω） for （FrCED）.

Henceˉx（t） =ˉu（t）， t∈I， i.eˉu（t） is an optimal solution.

To establish the converse duality theorem， we may write the problem （FrCED） in the form following form：

Maximize

Considerθ（r，t，x（t），x（t），¨x（t），y （t），y （t），z （t），z （t），w（t）） as defining a mapping where Q ： R+×X×Y×Z×WU， and Y and Z are spaces of differentiable functions y and z， W is the space of piecewise smooth functions w and R+is the set of nonnegative real numbers and U is a Banach space.

In the proof the following converse duality theorem， we have some restriction on the equality constraintθ（.） = 0 that appears in the dual （FrCED）. If suffices if the Frechet derivative Q！= [Qγ，Qx，Qy，Qz，Qw] has weak （*） closed range [4].

REFERENCES

[1] Weir， T.， & Mond， B. （1986）. Sufficient Fritz John optimality condition and duality for nonlinear programming problems. Opsearch， 23（3）， 129141.

[2] Hanson， M. A. （1964）. Bounds for functionally convex optimal control problems. J. Math. Anal. Appl.， 8， 8489.

[3] Mond， B.， & Hanson， M. A. （1967）. Duality for variational problems. J. Math. Anal. Appl.， 18， 355364.

[4] Chandra， S.， Craven， B. D.， & Husain， I. （1985）. A class of nondifferentiable continuous programming problems. 107（1）， 122131.

[5] Bector， C. R.， Chandra， S.， & Husain， I. （1985）. Generalized concavity and nondifferentiable continuous programming duality. Research Report # 85-7， Faculty of administrative studies， The University of Manitoba， Winnipeg， Canada R3T 2N2.

[6] Husain， I.， & Srivastav， S. K. （2012）. Fritz John Duality in the presence of equality and inequality constraints. Applied Mathematics， 3， 10231028.