My Math Forum  

Go Back   My Math Forum > High School Math Forum > Algebra

Algebra Pre-Algebra and Basic Algebra Math Forum


Thanks Tree2Thanks
  • 1 Post By JeffM1
  • 1 Post By SDK
Reply
 
LinkBack Thread Tools Display Modes
July 27th, 2018, 03:59 AM   #1
Senior Member
 
Joined: Apr 2014
From: Glasgow

Posts: 2,132
Thanks: 717

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 04:06 AM.
Benit13 is offline  
 
July 27th, 2018, 06:53 AM   #2
Senior Member
 
Joined: May 2016
From: USA

Posts: 1,206
Thanks: 494

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
JeffM1 is offline  
July 27th, 2018, 08:11 AM   #3
SDK
Senior Member
 
Joined: Sep 2016
From: USA

Posts: 520
Thanks: 293

Math Focus: Dynamical systems, analytic function theory, numerics
You are correct and the book is wrong.
Thanks from Benit13
SDK is offline  
Reply

  My Math Forum > High School Math Forum > Algebra

Tags
optimization, problem



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Optimization Problem codyrawlins Calculus 4 December 8th, 2017 10:23 PM
Optimization problem PhizKid Calculus 1 December 13th, 2012 05:04 AM
Optimization problem help 99rock99 Calculus 1 May 10th, 2012 08:10 PM
Optimization Problem .. Please help fantom2012 Calculus 6 April 25th, 2012 11:41 PM
Optimization Problem ryan.carmody Calculus 1 November 25th, 2007 02:16 PM





Copyright © 2018 My Math Forum. All rights reserved.