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July 27th, 2018, 02:59 AM  #1 
Senior Member Joined: Apr 2014 From: Glasgow Posts: 2,150 Thanks: 730 Math Focus: Physics, mathematical modelling, numerical and computational solutions  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 KarushKuhnTacker (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} $ Last edited by Benit13; July 27th, 2018 at 03:06 AM. 
July 27th, 2018, 05:53 AM  #2 
Senior Member Joined: May 2016 From: USA Posts: 1,310 Thanks: 551 
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.

July 27th, 2018, 07:11 AM  #3 
Senior Member Joined: Sep 2016 From: USA Posts: 598 Thanks: 366 Math Focus: Dynamical systems, analytic function theory, numerics 
You are correct and the book is wrong.


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