Constraint Minimization

Problem Statement

Minimize the following function $5-(x-2)^2 -2(y-1)^2$ subject to the following constraint $x + 4y = 3$

Solution

Lagrangian of the problem, $L(x,y, \lambda) = 5 - (x-2)^2 - 2(y-1)^2 + \lambda (3 - x-4y)$

$\frac{\partial L}{\partial x} = -2(x-2) - \lambda$

$\frac{\partial L}{\partial y} = -2(y-1) -4 \lambda$

$\frac{\partial L}{\partial \lambda} = 3 - x - 4y$

Putting the last 3 equations to 0, we get:

$\lambda = 4 - 2x$

$-2y + 2 - 4(4- 2x) = 0$

=> $-2y + 2 - 16 + 8x = 0$

=> $8x - 2y = 14$

=> $4x - y = 7$

=> $y = 4x - 7$

Since, $x + 4y = 3$ from last equation, we get:

$x + 4(4x - 7) = 3$

=> $x + 16x - 28 = 3$

=> $17x = 31$

=> $x = \frac{31}{17}$

Thus, $y = 4x - 7 = \frac{5}{17}$

Now, we have to see if the point we found is a minima or not. To check this, we see if the function is convex or concave.

Evaluating Hessian matrix:

$H(x,y)= \left[ {\begin{array}{cc} -2 & 0 \\ 0 & -4 \\ \end{array} } \right] $

Since $H(x,y)$ is negative definite, the given point is a maxima. This means that no minimum value exists for the function subject to the given constraint.