Oct 17

Lagrange method: sufficient conditions

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 f(x,y) subject to g(x,y)=0.

Everywhere we impose the implicit function existence condition:

(2) \frac{\partial g}{\partial y}\neq 0.

Differentiation of the restriction gives

(3) \frac{\partial g}{\partial x}+\frac{\partial g}{\partial y}y^\prime(x)=0.

Let (x,y) be an extremum point for (1). Then, as we proved, there exists \lambda such that the Lagrangian L(x,y,\lambda )=f(x,y)+\lambda g(x,y) satisfies FOC's:

(4) \frac{\partial L}{\partial x}=0, \frac{\partial L}{\partial y}=0, \frac{\partial L}{\partial \lambda }=0.

Also we need the function \phi (x)=f(x,y(x)) with the constraint built into it.

Heading to sufficient conditions

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

(5) \frac{d^2\phi }{dx^2}=\frac{\partial^2f}{\partial x^2}+2\frac{\partial^2f}{\partial x\partial y}y^\prime(x)+\frac{\partial^2f}{\partial y^2}[y^\prime(x)]^2+\frac{\partial f}{\partial y}y^{\prime\prime}(x).

Differentiating (3) once again gives

(6) \frac{\partial^2g}{\partial x^2}+2\frac{\partial^2g}{\partial x\partial y}y^\prime(x)+\frac{\partial^2g}{\partial y^2}[y^\prime(x)]^2+\frac{\partial g}{\partial y}y^{\prime \prime}(x)=0.

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

(7) \frac{d^2\phi }{dx^2}=\frac{\partial^2L}{\partial x^2}+2\frac{\partial^2L}{\partial x\partial y}y^\prime(x)+\frac{\partial^2L}{\partial y^2}[y^\prime(x)]^2+\frac{\partial L}{\partial y}y^{\prime\prime}(x).

Here \frac{\partial L}{\partial y}y^{\prime\prime}(x)=0 because of (4).

D^2L=\left(\begin{array}{cc}  \frac{\partial^2L}{\partial x^2}&\frac{\partial^2L}{\partial x\partial y}\\  \frac{\partial^2L}{\partial x\partial y}&\frac{\partial^2L}{\partial y^2}\end{array}  \right),\ Y=\left(\begin{array}{c}1\\y^\prime(x)\end{array}\right).

Then (7) rewrites as

(8) \frac{d^2\phi}{dx^2}=Y^TD^2LY.

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

Rough sufficient condition. If we require the Hessian to be positive definite, then h^TD^2Lh>0 for any h\neq 0 and, in particular, for h=Y. Thus, positive definiteness of D^2L, together with the FOC (3), will be sufficient for \phi to have a minimum.

Refined sufficient condition. We can relax the condition by reducing the set of h on which h^TD^2Lh should be positive. Note from (3) that Y belongs to the set Z=\{(a_1,a_2):\frac{\partial g}{\partial x}a_1+\frac{\partial g}{\partial y}a_2=0\}. Using (2), for (a_1,a_2)\in Z we can write a_2=(-\frac{\partial g}{\partial x}/\frac{\partial g}{\partial y  })a_1. This means that Z is a straight line. Requiring h^TD^2Lh>0 for any nonzero h\in Z we have positivity of (8) for h=Y. 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 h^TD^2Lh>0 for any nonzero h\in Z, then that point is a minimum point. b) If at that point h^TD^2Lh<0 for any nonzero h\in Z, then that point is a maximum point.

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