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Re: Question on two-stage optimization of polynomials:

On 11/10/2011 05:55 AM, lao kn wrote:
> I need to solve an optimization problem. Please offer your insights.
> Suppose there are two functions (my case is more complicated than this
> example).
> f(x,y)=x-y, 0<x<1, 0<y<1
> g(x,y)=x*y, 0<x<1, 0<y<1
> The goal is to get the maximum values for f(x,y) and g(x,y). However,
> it is not simultaneous, but sequential. In the first stage, only x can
> be chosen to maximize f(x,y); in the second stage only y can be chosen
> to maximize g(x,y).
> To solve this problem, I should start from the second stage, say, let
> x varies from 0 to 1, and get each y that maximizes g(x,y). or in
> another word, derive the best response function y*=v(x). Then I can
> put the response function back to f(x,y) and solve for the x that
> maximizes f(x,y).
> An easy way is to get the reponse function is to differentiate g(x,y)
> respect to y, and the set it to zero: df/dy=0, and the response
> function is at hand. However, my case is too complex to get the
> differentiation. Therefore I need to solve the problem numerically.
> Therefore, what I want is a function or a curve that can let x varies
> from 0 to 1 and see the best response value of y. Even if I can not
> get a response function, I hope I can get a response curve drawn by
> Mathematica.
> Can someone help me out this trouble?
> Many thanks

Possibly something like the code below will do what you want. We regard 
g as a function of one variable, y. We use f to solve for x as a 
function of y.

Clear[f, g, y, x]

f[x_, y_] := x + y
g[y_] := x[y]*y;
x[y_?NumericQ] :=
  Module[{xx}, xx /. Last[NMaximize[{f[xx, y], 0 <= xx <= 1}, xx]]]

In[172]:= NMaximize[{g[y], 0 <= y <= 1}, y]
Out[172]= {1., {y -> 1.}}

Daniel Lichtblau
Wolfram Research

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