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Benit13 July 27th, 2018 02:59 AM

Optimization problem
 
I have the following optimization problem:

$\displaystyle \textrm{Minimize }f(x) + g(y)$
$\displaystyle x, y$
$\displaystyle \textrm{Subject to } h(x,y) = x + y - k = 0$

where the cost functions are quadratic:

$\displaystyle f(x) = a_1 + b_1x + \frac{c_1}{2} x^2$
$\displaystyle g(y) = a_2 + b_2y + \frac{c_2}{2} y^2$

The Lagrangian is defined as

$\displaystyle \mathcal{L}(x, y, \lambda) = f(x) + g(y) - \lambda h(x,y)$

Therefore, the Lagrangian of this problem is

$\displaystyle \mathcal{L}(x, y, \lambda) = a_1 + b_1x + \frac{c_1}{2} x^2 + a_2 + b_2y + \frac{c_2}{2} y^2- \lambda(x + y - k)$

The book that I have says that the Karush-Kuhn-Tacker (KKT) optimality conditions for this problem are

$\displaystyle \frac{\partial \mathcal{L}}{\partial x} = c_1 x - \lambda = 0$
$\displaystyle \frac{\partial \mathcal{L}}{\partial y} = c_2 y - \lambda = 0$
$\displaystyle \frac{\partial \mathcal{L}}{\partial \lambda} = -(x + y - k) = 0$

but if these are just partial derivatives of the Lagrangian, then what happened to $\displaystyle b_1$ and $\displaystyle b_2$? Should they instead be

$\displaystyle \frac{\partial \mathcal{L}}{\partial x} = b_1 + c_1 x - \lambda = 0$
$\displaystyle \frac{\partial \mathcal{L}}{\partial y} = b_2 + c_2 y - \lambda = 0$
$\displaystyle \frac{\partial \mathcal{L}}{\partial \lambda} = -(x + y - k) = 0$

The book states that the solutions are:

$\displaystyle x = \frac{c_2}{c_1 + c_2} k$
$\displaystyle y = \frac{c_1}{c_1 + c_2} k$
$\displaystyle \lambda = \frac{c_1 c_2}{c_1 + c_2} k$

If I include $\displaystyle b_1$ and $\displaystyle b_2$, I get the following solutions instead:

$\displaystyle x = \frac{b_2 - b_1 + c_2 k}{c_1 + c_2}$
$\displaystyle y = \frac{b_1 - b_2 + c_1 k}{c_1 + c_2}$
$\displaystyle \lambda = \frac{b_1 c_c + b_2 c_2 + c_1 c_2 k}{c_1 + c_2} $

JeffM1 July 27th, 2018 05:53 AM

This is truly a superficial response, but the book's answer (if correct) seems to imply that the b parameters are known (somehow) to equal zero (or to be negligible in practice). Since this model seems to come from economics, implicit assumptions may be hidden almost anywhere.

SDK July 27th, 2018 07:11 AM

You are correct and the book is wrong.


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