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 July 27th, 2018, 02:59 AM #1 Senior Member   Joined: Apr 2014 From: Glasgow Posts: 2,157 Thanks: 732 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 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: $\displaystyle x = \frac{b_2 - b_1 + c_2 k}{c_1 + c_2}$ $\displaystyle y = \frac{b_1 - b_2 + c_1 k}{c_1 + c_2}$ $\displaystyle \lambda = \frac{b_1 c_c + b_2 c_2 + c_1 c_2 k}{c_1 + c_2}$ 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. Thanks from Benit13 July 27th, 2018, 07:11 AM #3 Senior Member   Joined: Sep 2016 From: USA Posts: 635 Thanks: 401 Math Focus: Dynamical systems, analytic function theory, numerics You are correct and the book is wrong. Thanks from Benit13 Tags optimization, problem Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post codyrawlins Calculus 4 December 8th, 2017 09:23 PM PhizKid Calculus 1 December 13th, 2012 04:04 AM 99rock99 Calculus 1 May 10th, 2012 07:10 PM fantom2012 Calculus 6 April 25th, 2012 10:41 PM ryan.carmody Calculus 1 November 25th, 2007 01:16 PM

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