Hierarchical optimization

I am looking to find a point of a function f(x,y)f(x,y)f(x,y) that minimizes fff with respect to xxx and at the same time fulfills x=yx=yx=y (but the point is not necessarily a minimum with respect to yyy). I would consider this a hierarchical optimization. Here a simple example:

FindRoot[y == x /. NMinimize[(x – y^2)^2, x][[2]], {y, 1}]

This code does not work. What I intend to do is find yyy such that y==xy==x and xx minimizes f(x,y)=(x−y2)2f(x,y)=(x-y^2)^2. The solution would be y=1y=1. Is there a syntactically correct way to do this in Mathematica?

Edit: The example was poorly chosen because ff has a minimum with respect to yy as well at x=y=1x=y=1. Here is a modified example:

f[x_, y_] := (x – y^2)^2 + Exp[y]

This is the result I would like to obtain:

Block[{y = 1}, NMinimize[f[x, y], x][[2]]]
(* {x -> 1.} *)

However, minimizing f[x,x]f[x,x] leads to a wrong solution:

NMinimize[{f[x, x], x > 0.5}, x]
(* {x -> 0.5} *)

With pen-and-paper math, the solution I am aiming for reads

Minimize with ff respect to xx yields x=y2x=y^2.
Solve x=yx=y gives y=y2y=y^2 or y=0,1y=0,1.

I am looking for a numerical method to obtain the same result.

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Only use NMinimize and replace y with x
– Jack LaVigne
Oct 21 at 14:51

  

 

For my first example this approach worked. I have modified the example to better illustrate my problem.
– Felix
Oct 21 at 15:41

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3 Answers
3

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f[x_, y_] := (x – y^2)^2 + Exp[y]
g[y_?NumericQ] := x /. NMinimize[f[x, y], x][[2]]
FindRoot[g[y] == y , {y, 1.25}]

{y->1.}

  

 

Yes, this is exactly what I was looking for. Why does it not work when writing it in one line, i.e., without the auxiliary function g?
– Felix
Oct 21 at 18:41

1

 

the g function ensures that NMinimize is only evaluated with a numeric y value.
– george2079
Oct 21 at 18:46

You can find the minimum by taking the derivative and setting equal to zero.

f[x_, y_] := (x – y^2)^2 + Exp[y];
sol = Solve[D[f[x, y], x] == 0, x]

This gives {{x -> y^2}}, and you now want to solve for when both sides of this are equal:

Solve[y == First@(x /. sol), y]

{{y -> 0}, {y -> 1}}

  

 

Unfortunately, this analytical approach does not work for my actual problem, because Solve is limited to very simple (polynominal) functions. I need a numerical method.
– Felix
Oct 21 at 18:20

Starting with Bill S’s answer, you can combine the steps using Solve

Solve[{D[f[x, y], x] == 0, x == y}, {x, y}]
(* {{x -> 0, y -> 0}, {x -> 1, y -> 1}} *)

or use Reduce which tends to be a bit more robust

Reduce[2 (x – y^2) == 0 && x == y, {x, y}]
(* (x == 0 || x == 1) && y == x *)

and if you need a numerical solution

NSolve[{D[f[x, y], x] == 0, x == y}, {x, y}, Reals]
(* {{x -> 1., y -> 1.}, {x -> 0., y -> 0.}} *)

  

 

+1 for a numerical solution that is also much faster than george’s version. George’s solution, however, works with more complicated functions, where NMinimize is required to find the minimum (solving for derivative == 0 often fails when the functions involved are non-trivial).
– Felix
Oct 22 at 9:56