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Exponent optimization with constrains

Posted 9 years ago

Hello, I'm trying to find an optimization for each player (each one has a utilitiy function) with constraints. the problem has an exponent factor. I don't know how to construct the problem in mathematica and to find a feasible solution for each player (or potentialy an equilibrium).

Thanks!

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POSTED BY: yossi sh
5 Replies
Posted 9 years ago

From the initial post I was guessing that you needed to maximize both of the items.

In[1]:= s = 95; u = 0.1; x = 100; k = 0.01; Subscript[c, A] = (Subscript[p, A](s(1 + u) - x)) + k; Subscript[c, B] = (Subscript[p, B](s(1 + u) - x)) - k; Subscript[Z, A] = (1 - (v - Subscript[c, A])/(Subscript[c, B] - Subscript[c, A]))^[Beta] (v - Subscript[c, A]); Subscript[Z, B] = (1 - (v - Subscript[c, B])/(Subscript[c, B] - Subscript[c, A]))^[Alpha] (Subscript[c, B] - v); Maximize[{Subscript[Z, A], Subscript[c, A] <= v <= Subscript[c, B] && 0 <= Subscript[p, A] <= 1 && 0 <= Subscript[p, B] <= 1}, {v, [Alpha], [Beta], Subscript[p, A],Subscript[p, B]}]

During evaluation of In[1]:=NMaximize::nrnum:The function value -0.113773+0.104822 I is not a real number at {v,[Alpha],[Beta],Subscript[p, A],Subscript[p, B]} = {1.24375,0.162408,-0.0367399,0.0548333,0.081051}. >>

Out[6]= {0.863531, {v -> 0.754547, [Alpha] -> -0.523666, [Beta] -> -0.734744, Subscript[p, A] -> 0.0000800015, Subscript[p, B] -> 0.906934}}

In[7]:= Maximize[{Subscript[Z, B], Subscript[c, A] <= v <= Subscript[c, B] && 0 <= Subscript[p, A] <= 1 && 0 <= Subscript[p, B] <= 1}, {v, [Alpha], [Beta], Subscript[p, A],Subscript[p, B]}]

During evaluation of In[7]:=NMaximize::nrnum:The function value 1.613510^8-4.6536810^7 I is not a real number at {v,[Alpha],[Beta],Subscript[p, A],Subscript[p, B]} = {0.377249,18.8636,-19.7831,0.162586,0.150344}. >>

Out[7]= {2.99746*10^7, {v -> 0.454771, [Alpha] -> 16.1621, [Beta] -> -15.78, Subscript[p, A] -> 0.187606, Subscript[p, B] -> 0.2869}}

POSTED BY: Bill Simpson
Posted 9 years ago

I think that Za and Zb should be maximized seperately. Is this what you've wrote?

POSTED BY: yossi sh
Posted 9 years ago
In[1]:= s = 95; u = 0.1; x = 100; k = 0.01;
Subscript[c, A] = (Subscript[p, A]*(s*(1 + u) - x)) + k;
Subscript[c, B] = (Subscript[p, B]*(s*(1 + u) - x)) - k;
Subscript[Z, A] = (1-(v-Subscript[c,A])/(Subscript[c,B]-Subscript[c,A]))^\[Beta](v-Subscript[c,A]);
Subscript[Z, B] = (1-(v-Subscript[c,B])/(Subscript[c,B]-Subscript[c,A]))^\[Alpha](Subscript[c,B]-v);
Maximize[{Norm[Subscript[Z,A]]+Norm[Subscript[Z,B]], Subscript[c,A]<=v<=Subscript[c,B] && 
   0<=Subscript[p,A]<=1 && 0<=Subscript[p,B]<= 1}, {v, \[Alpha], \[Beta], Subscript[p,A], Subscript[p,B]}]

During evaluation of In[1]:=NMaximize::cvdiv:Failed to converge to a solution. The function may be unbounded.

Out[9]= {3.97046*10^15, {v -> 0.101697, \[Alpha] -> 53.8285, \[Beta] -> -34.2923, Subscript[p, A] -> 0.0203771, 
  Subscript[p, B] -> 0.0799841}}

You may want to look at the values of each part of ZA and ZB to determine whether more constraints are needed or whether there is something about the way that I have done this which is incorrect.

POSTED BY: Bill Simpson
Posted 9 years ago

0<=Pa<=1, 0<=Pb<=1

POSTED BY: yossi sh
Posted 9 years ago

Can you provide any additional constraint information?

In[1]:= s = 95;
u = 0.1;
x = 100;
k = 0.01;
Subscript[c, A] = (Subscript[p, A]*(s*(1 + u) - x)) + k;
Subscript[c, B] = (Subscript[p, B]*(s*(1 + u) - x)) - k;
Subscript[Z, A]=(1-(v-Subscript[c,A])/(Subscript[c,B]-Subscript[c,A]))^\[Beta] (v-Subscript[c,A]) ;
Subscript[Z, B]=(1-(v-Subscript[c,B])/(Subscript[c,B]-Subscript[c,A]))^\[Alpha] (Subscript[c,B]-v);
Maximize[{Norm[Subscript[Z, A]] + Norm[Subscript[Z, B]], Subscript[c, A] <= v <= Subscript[c, B]},
   {v, \[Alpha], \[Beta], Subscript[p, A], Subscript[p, B]}]

During evaluation of In[61]:=NMaximize::cvdiv:Failed to converge to a solution. The function may be unbounded.

Out[68]= {1.945590892669753*10^514, {v -> -48.2786, \[Alpha] -> 56.0364, \[Beta] -> -102.072,
   Subscript[p, A] -> -52.0972, Subscript[p, B] -> -10.7259}}
POSTED BY: Bill Simpson
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