## Lagrange method: sufficient conditions

From what we know about unconstrained optimization, we expect that somehow the matrix of second derivatives should play a role. To get there, we need to differentiate twice the objective function with the constraint incorporated.

## Summary on necessary condition

(1) The problem is to maximize subject to

Everywhere we impose the implicit function existence condition:

(2)

Differentiation of the restriction gives

(3)

Let be an extremum point for (1). Then, as we proved, there exists such that the Lagrangian satisfies FOC's:

(4)

Also we need the function with the constraint built into it.

## Heading to sufficient conditions

We need to check the sign of the second derivative of :

(5)

Differentiating (3) once again gives

(6)

Since we need to obtain the Lagrangian, let us multiply (6) by and add the result to (5):

(7)

Here because of (4).

Denote

Then (7) rewrites as

(8)

This is a quadratic form of the Hessian of (no differentiation with respect to ).

**Rough sufficient condition**. If we require the Hessian to be positive definite, then for any and, in particular, for . Thus, positive definiteness of , together with the FOC (3), will be sufficient for to have a minimum.

**Refined sufficient condition**. We can relax the condition by reducing the set of on which should be positive. Note from (3) that belongs to the set . Using (2), for we can write This means that is a straight line. Requiring for any nonzero we have positivity of (8) for We summarize our findings as follows:

**Theorem**. Assume the implicit function existence condition and consider a critical point for the Lagrangian (that satisfies FOC's). a) If at that point for any nonzero , then that point is a minimum point. b) If at that point for any nonzero , then that point is a maximum point.

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