Fritz John Duality in the Presence of Equality and Inequality Constraints

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.

• Second-Order Invexity; Second-Order Pseudoinvexity; Second-Order Quasi-Invexity; Nonlinear Programming; Fritz John Type Dual

1. Introduction

Consider the following mathematical programming problems.

(NP): Minimize

Subject to

(NEP): Minimize

Subject to

where, and are differentiable functions. The best-known necessary optimality conditions for the mathematical programming problem (NP) and (NEP) are Fritz John necessary optimality conditions and Karush-Kuhn-Tucker type optimality conditions. The Fritz John type [1] optimality condition which predates the Karush-Kuhn-Tucker type optimality conditions by a few years are more general in a sense. In order for Karush-Kuhn-Tucker type optimality conditions to hold, a constraint qualification or regularity condition on the constraint is required. On the other hand, no such constraint qualification is needed for Fritz John optimality conditions to hold.

Fritz John [2] established the following optimality conditions for (NP):

Proposition 1. (Fritz John type necessary conditions). If is an optimal solution of (NP), then there exist and a vector such that

Using these optimality conditions, Weir and Mond [3] formulated the for Fritz John type dual to (NP) and established usual duality theorems, this eliminating the requirement of a constraint qualification:

: Maximize

Subject to

.

Originally, Fritz John derived his optimality condition for the case of inequality constraint alone. If equality constraint are present in a mathematical programming problem and they are converted into two inequality constraints, then the Fritz John optimality conditions become useless because every feasible point satisfying them. Later Mangasarian and Fromovitz [4] derived necessary optimality condition for (NEP) without replacing an equality constraint by two inequalities and hence making it possible to handle equalities and inequalities together as many realistic problems contain both equality and inequality constraint. Mangasarian and Fromovitz [4] established the following Fritz John type optimality condition given in the following propositions:

Proposition 2.(Generalized Fritz John necessary optimality Conditions [4]):

If is an optimal solution of (NEP), then there exist, and such that

2. Sufficiency of Fritz John Optimality Conditions

Before proceeding to the main results of our analysis we give the following definitions which are required for their validation.

1) The function is strictly pseudoconvex on for all

Equivalently

2) For and is said to be semi-strictly pseudoconvex if is strictly pseudoconvex for all

Theorem 1. (Sufficient Optimality Conditions):

Assume that

1) is pseudoconvex2) is semi strictly pseudoconvex and 3) is quasiconvexIf there exist, , and such that (3)-(8) are satisfied, then is an optimal solution of (NEP).

Proof: Suppose is not optimal, i.e., and then there exists Such that

Since is pseudoconvex, this implies

and

with strict-inequality in the above if

Since is feasible for (NEP) we have

Because of semi strict pseudoconvexity of, This implies

With strict inequality with,.

Also

Because of quasi-convexity of at,

Combining (7), (8) and (9), we have

Contradicting (3). Hence is an optimal solution of (NEP).

3. Fritz John Type Duality

We propose the following dual (F_rED) to (NEP), using Fritz John optimality conditions stated in the preceding section instead of Karush-Kuhn-Tucker conditions [5,6] and established duality results, thus the requirement of a constraint qualification [4] is eliminated:

Dual Problem:

(F_rED): Maximize

Subject to

Theorem 2. (Weak Duality):Assume that

: x is feasible for (NEP) and is feasible for.

: For all feasible, is pseudoconvex, is semi-strictly pseudoconvex and is quasiconvex.

Then

Proof: Suppose this, because of pseudoconvexity of yields, Multiplying this, by We have

With strict inequality in (15) if

From the Constraints of and, we have

which by semi-strictly pseudoconvexity of implies

with strict inequality in (16) if

As earlier

This along with quasiconvexity of implies

Combining (15), (16), (17), we have

Contradicting

Hence

This implies.

Theorem 3. (Strong Duality):

If is an optimal solution of then there exist, and such that is feasible for and the corresponding values of and are equal. If, also f is pseudoconvex, is semi-strictly pseudoconvex and is quasi-convex, then is an optimal solution of.

Proof: Since is an optimal solution of, by Proposition 2. There exist, and such that

This implies is feasible for. Equality of objective function of and is abovious optimality follows, in view of the hypothesis of the theorem1.

Theorem 4. (Strict Converse Duality): Assume that

1) is strictly pseudoconvex, is semistrictly pseudoconvex and as is quasiconvex and 2) The problem has an optimal solution.

If is an optimal solution of, Then i.e. is an optimal solution of.

Proof: We assume that and exhibit a contradiction, it follows from Proposition 2 that there exist, and such that is optimal solution of, since is also an optimal solution for, It follows that

by strict pseudoconvexity of we have

Also from the constraints of and we have.

By the semi strictly convexity of, this implies

with strict inequality in the above, if

Also which by quasi-convexity of at, implies

Combining (18), (19), and (20), we have

which contradicts

Hence is an optimal solution.

Theorem 5. (Converse Duality): If is an optimal solution of. Assume that

: is pseudoconvex, is semi strictlypseudoconvex and is quasiconvex.

: Hessian matrix is positive or negative definite, and

: the set is linearly independent, and Then is an optimal solution of.

Proof: By Preposition 2, there exist, , , and such that

Multiplying (23) by y ≥ 0 and using (25) and (28), we obtain

Multiplying (24) by and we have

Multiplying equality constraint of by and using (31) and (32) We have

Multiplying (21) by and using (31) and (32), we have

Multiplying the above equation by r and using (33), we have

This because of hypothesis (A₂) implies rθ = 0. In view of (A₃) the equality constraint of implies r ≠ 0, i.e., r > 0.consequently θ = 0.

Multiplying (21) by r and using θ = 0, we have

Using the equality constraint (10) in the above, we have

This reduces to

By the linear independence hypothesis (A₃). this implies

and

Now if τ = 0, then from above, we have ϕ = 0, ψ = 0 and from (22) and (23), We have ξ = 0, η = 0, consequently we have contradicting to (30).

Hence t > 0, ϕ > 0, and ψ > 0.

Using in (23) and (24), we have

,

This implies and

Thus is feasible for and the objective functions of and are equal in their formulations. Under, the state generalized Convexity, Theorem 1 implies that, is an optimal solution of.

4. Generalized Fritz John Duality

Let and with and

. and with, and. Let and The following is the generalized Fritz John type dual to.

Maximize

Subject to

Theorem 6: If is pseudoconvex, is semi-strictly pseudoconvex, and is quasiconvexThen

Proof: Let  be feasible for  and  feasible for. Suppose  This by pseudoconvexity of  yields

with strict inequality in (34) if

From the constraint of and, we have

Which because of semistrictly pseudoconvexity of implies

with strict inequality in (36) if some

Also

And

Which by quasiconvex of and

respectively imply

and

combining (34), (35), (36) and above equation we have

contradicting the equality constraint of. Hence

Implying

Theorem 7. (Strong Duality):

If is an optimal solution of and there exist, and such that is feasible for and the corresponding value of and are equal. If, the hypotheses of Theorem 1 hold, then is an optimal solution of.

Proof: By Proposition 2, there exist, and such that

Since, and feasibility of for is obvious. Optimality follows, give the pseudoconvexity of

and semi-strict pseudoconvexity of quasiconvexity of and quasiconvexity of from Theorem 1.

Theorem 8: (Mangasarian [4] Type Strict Converse Duality): Assume that

is strictly pseudoconvex,

is semi-strictly pseudoconvex and

and are quasiconvex.

is an optimal solution of.

If is an optimal solution of then i.e. is an optimal solution of.

Proof: Assume that and exhibit a contradiction. Since is an optimal solution of, by Proposition 2, it implies that there exist, and such that is an optimal solution of.

Since is an optimal solution for, it follows that

This, in view of strict pseudoconvexity of implies

From the constraint of and, we have

and

The inequality (38), in view of semi-strict pseudo convexity of implies

with strict inequality in (40) if.

By quasiconvexity of (38) implies

The inequality (39), because of quasiconvexity of yields,

Combining (37), (40), (41) and (42), we have

which contradicts the feasibility of for . Hence

Theorem 9 (Converse Duality): Let

be an optimal solution of.

be pseudoconvex, semistrictly pseudoconvex, quasiconvex.

The Hessian matrix

is positive or negative definite, and

The set

is linearly independent. Then is feasible for.

Proof: By Proposition 2, there exist , and such that

Multiplying (45) and (46) by and respectively and using (47) and (48), we have

Multiplying (44) by r, we have

Multiplying (43) by and using (53), (54) and (53), we have

By positive or negative definite and by hypothesis, we have

In view of, equality constraint of implies that Hence using we have

which in view of the hypothesis gives ,. From (44) and (45), we have and consequently we have

Contradicting Fritz John Condition (51). Hence since The Equations (45) and (46), implies

Thus is feasible for and optimality follows as earlier.

5. Conclusion

In this exposition, we have formulated a dual and generalized dual by Fritz John optimality conditions instead of the Karush-Kuhn-Tucker optimality conditions. Consequently no constraint qualification is required and hence such formulations enjoy computational advantage over those formulated by using Karush-Kuhn-Tucker. The problems of these results can be revisited in multiobjective and dynamic setting.

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.