       Re: NMinimize--problem with a min-max problem

• To: mathgroup at smc.vnet.net
• Subject: [mg55804] Re: [mg55753] NMinimize--problem with a min-max problem
• From: Bob Hanlon <hanlonr at cox.net>
• Date: Wed, 6 Apr 2005 03:12:05 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```expr=(Log[x]+4/3 z Log[1-x])/(2 (3/4+z));

solnx=Solve[D[expr,x]==0,x][]

{x -> 3/(4*z + 3)}

solnz=Solve[D[expr/.solnx,z]==0,z][]

{z -> 3/4}

solnx=solnx/.solnz

{x -> 1/2}

expr/.solnx/.solnz

-((2*Log)/3)

%//N

-0.46209812037329684

Clear[fun1];
fun1[z_?NumericQ] :=
NMaximize[{(Log[x]+4/3 z Log[1-x])/(2 (3/4+z)),
0<x<1},{x}][];

Off[NMaximize::nnum];

NMinimize[fun1[z],z]

{-0.4620981203732968, {z -> 0.7499999786528403}}

Bob Hanlon

>
> From: "David" <isolanoster at gmail.com>
To: mathgroup at smc.vnet.net
> Date: 2005/04/05 Tue AM 03:21:55 EDT
> To: mathgroup at smc.vnet.net
> Subject: [mg55804] [mg55753] NMinimize--problem with a min-max problem
>
> Hello,
>
> I'm trying to solve numerically the following problem:
>
> min{z}max{y,x} f{x,y,z}
>
> I first do the maximization:
>
> fun1[z_] := NMaximize[{(
>   Log[x] + 4/3 z Log[
>       y])/(2 (3/4 + z)), x > 0, y > 0, x + y == 1}, {x, y}][]
>
> I can plot fun1[z] (the solution is at z=.75,x=y=1/2) however NMinimize
> does accept the definition of fun1[z] as:
>
> NMinimize[fun1[z], z]
>
> yields error.
>
> Any suggestion?
>