numerics - Combined numerical minimization and maximization

I want to numerically calculate the maximum of a function defined by the minimization of another function, like the following:

NMaximize[b/.NMinimize[(a^2+b)^2,{b}][[2]],{a}]

Obviously the intended result (for this simple test function) would be:

{0., {a->0.}}

because the inner minimization would give b=-a^2 for any value of a, and maximizing that gives 0 for a=0.

However I get the error message NMinimize::nnum: "The function value (-0.829053+a^2)^2 is not a number at {b} = {-0.829053}." and the result NMaximize[b/.{b},{a}]. I figured that this is due to premature evaluation of the argument (i.e. before a got anumerical value), therefore I tried to wrap either the whole first argument of just the NMinimize call in Unevaluated, but neither helped.

So my question is: How can I do this combined numerical optimization?

Answer

This is a rather common issue that comes up with many numerical functions (FindRoot, NIntegrate, FindMaximum, NMaximize, etc.) It is also explained in this Wolfram Knowledge Base article. Sometimes you want to pass these functions an expression that has a symbolic parameter, and compute the result for different values of that parameter.

Example:

fun[a_] := Block[{b}, b /. NMinimize[(a^2 + b)^2, {b}][[2]]]

This will work nicely if you call it with a numeric argument: fun[3]. But it will cause an error in NMinimize if you call it with a symbolic parameter: fun[a] (for obvious reasons).

The solution is:

Clear[fun]
fun[a_?NumericQ] := Block[{b}, b /. NMinimize[(a^2 + b)^2, {b}][[2]]]

NMaximize[fun[a], {a}]

(Be sure to evaluate Clear[...] to get rid of the previous definition of fun!)

This ensures that fun will only evaluate for numerical arguments, i.e. fun[a] won't evaluate inside NMaximize before NMaximize actually substitutes a number for a.

And this is also the answer to your specific question: make the inner NMinimize expression a separate function, and make sure it only evaluates for numerical arguments.

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Requested edit

An important related point is: how can we match only numerical quantities using a pattern? One might think of using _Real (as in the comment below). The problem with this is that it will only match numbers whose Head is Real. This excludes integers (such as 1,2,3), rationals (2/3, 4/5), constants (such as Pi or E), or expressions like Sqrt[2].

The only robust solution is using NumericQ[] (x_ ? NumericQ in a pattern). NumericQ will return True for anything that gives a number when N[] is applied to it.

There's another related function, NumberQ[], which gives True only for objects with Integer, Rational, Real or Complex, but not for constant or expressions (Pi or Sin[3]).