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The Kuhn-Tucker conditions have been used to derive many significant results in economics. However, thus far, their derivation has been a little bit troublesome. The author directly derives the Kuhn-Tucker conditions by applying a corollary of Farkas’s lemma under the Mangasarian-Fromovitz constraint qualification and shows the boundedness of Lagrange multipliers.

The Kuhn-Tucker conditions have been used to derive many significant results in economics, particularly in decision problems that occur in static situations, for instance, to show the existence of an equilibrium for a competitive economy (Negishi [

The Kuhn-Tucker conditions for the optimization problem with inequality and equality constraints have a comprehensive form that incorporates the method of Lagrange multipliers (introduced by Lagrange in 1788) in a natural way; therefore, the simple derivation of the Kuhn-Tucker conditions would shed light on the problem’s true nature.

In this paper, the Kuhn-Tucker conditions under the Mangasarian-Fromovitz constraint qualification are derived directly by applying a corollary of Farkas’s lemma without resorting to the Fritz John conditions and the boundedness of Lagrange multipliers is also shown.

The problem to be addressed is as follows:

where

Here, we should pay attention to the fact that the problem (P) naturally includes the optimization problem with equality constraints considered by Lagrange in the 18th century.

We postulate the following Mangasarian-Fromovitz constraint qualification (MF) (Mangasarian and Fromovitz [

(MF) For

Remarks

The linearly independent constraint qualification, which is usually assumed in practice, implies (MF) (see Nocedal and Wright [

(MF) is equal to the Cottle constraint qualification without the presence of equality constraints, and if the problem (P) is a concave program without equality constraints, the Slater constraint qualification implies the Cottle constraint qualification (Bazaraa and Shetty [

Finally, we recall the following result to the linear system including equalities for the sake of convenience.

Lemma 1. ([

(a)

or

(b)

but never both.

We now establish the main result, which differs from [

Theorem 1. Suppose that

and

Proof. At a local solution

which shows that

Then, for a local solution

does not hold, since, if so,

tradicts the local optimality of

Note that (MF) guarantees the existence of such

or, equivalently,

for

The rest part of the proof is as follows. From (3) we obtain

So, if

and if

Since

Example 1.

Consider the problem

with an optimal solution

Indeed, (MF) is valid for problems with a number of inequality constraints and admits feasible directions around

In this paper, the Kuhn-Tucker conditions under the Mangasarian-Fromovitz constraint qualification were derived directly by applying a corollary of Farkas’s lemma without resorting to the Fritz John conditions, or without introducing the Bouligand tangent cone, and the boundedness of Lagrange multipliers was also shown.

Considerable effort has been devoted to the generalization of Farkas’s lemma. However, what seems to be lacking is a discrete version of Farkas’s lemma under a mild condition; such a version would be theoretically meaningful and would be help solve the discrete optimization problems that emerge in the economics studying indivisible goods.