1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Optimization problem

  1. Aug 11, 2017 at 1:08 AM #1
    1. The problem statement, all variables and given/known data
    For positive x, y, z where ##x \le y \le z## such that x + y + z = 3, what is the maximum value of ##xy##?

    2. Relevant equations


    3. The attempt at a solution
    First, before I attempt a solution, isn't it the case that ##x=y=z=1##, since the only partition of 3 into three terms is 1+1+1? I feel like if this would the case the problem would be way too simple, which is why I'm asking.
     
  2. jcsd
  3. Aug 11, 2017 at 1:59 AM #2

    andrewkirk

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Only if x,y,z have to be integers. In optimisation problems, one usually assumes they can be any real numbers, unless advised otherwise. So here are three other possibilities:

    x=0.1, y=0.2, z=2.7
    x=0.5, y=1, z=1.5
    x=0.7, y=1.1, z=1.2

    The set of allowable points will be a solid polygon.
     
  4. Aug 11, 2017 at 2:41 AM #3

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    You have found the globally optimal solution over the region ##\{x >0,y \geq x,z \geq y, x+y+z=3 \}## but proving that this is the case may be tricky (because you have a non-convex optimzation problem, so in principle could have numerous local maxima). However, a bit of further re-writing shows this to not be the case: there is only one solution, and you have found it.

    First, re-write your final constraint as ##x+y+z \leq 3##; it will be satisfied as an equality at the maximum, (because for any point where the inequality is strict, we can move it out a bit to the boundary and keep the other constraints satisfied, while also increasing ##x y##).

    So now we want to maximize a non-convex function ##xy## but over a convex region. As is done in so-called Geometric Programming, we can change variables to ##x=e^u, y = e^v, z = e^w##. We want to minimize ##x^{-1} y^{-1}##, so our problem becomes to minimize ##f(u,v) = -u-v##, subject to constraints ##u-v \leq 0,## ## v-w \leq 0## and ##e^u + u^v + e^w \leq 3##. The function ##g(u,v,w) = e^u + u^v + e^w## is strictly convex, so the region ## g(u,v,w) \leq 3## is convex. Altogether, we have a convex programming in ##(u,v,w)##, and so any local constrained minimum (of ##f(u,v)##) is automatically a global constrained minimum. In other words, the solution is unique and can be found from the Karush-Kuhn-Tucker conditions.
     
    Last edited: Aug 11, 2017 at 3:12 AM
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Optimization problem
  1. Optimization problem (Replies: 2)

Loading...