Maximum of the minimum of two functions

I have two functions, each with 15 variables. How can I find maximum (minimum) of these functions? I know that minimum (maximum) of two functions is a multi-criteria function but I cannot find it. For example, assume two function are these:

$f(x,y,z)=x+2y-5z$

$g(x,y,z)=4x-y+9z$

• May look up FindMaxValue and FindMinValue in the documentation.
– kale
Nov 3, 2015 at 21:17
• no, those functions find maximum(minimum) of a function. Nov 3, 2015 at 21:23
• What do you mean by "...maximum... is a multi-criteria function"? Nov 3, 2015 at 21:33
• i say that minimum of two functions, is a function too,such that multi-criteria function and so on... but it is not a number. Nov 3, 2015 at 21:41
• I simply do not understand what you mean by "multi-criteria function." Can you give an example of a non multi-criteria function and an example of a multi-criteria function? Do you mean, for example, MinMax? Nov 3, 2015 at 22:28

One common approach for "maxmin" problems, i.e., of the form

maximimize (over $\{x_i\}$) $\min \{f_i(x_i)\}$,

is to introduce an additional variable, say $t$, and reformulate the problem as

maximize (over $t$ and $\{x_i\}$) $t$, subject to $f_i(x_i)\geq t, \forall i$.

Example:

f[x_, y_, z_] := x + 2 y - 5 z;
g[x_, y_, z_] := 4 x - y + 9 z;
FindMaximum[{t,
f[x, y, z] >= t && g[x, y, z] >= t && 0 <= x <= 0.1 &&
0 <= y <= 10 && 0 <= z <= 1}, {{t, 0}, {x, 0}, {y, 0}, {z, 0}}]

{4.63333, {t -> 4.63333, x -> 0.1, y -> 4.76667, z -> 1.}}

P.S.: I imposed additional (arbitrary) constraints on the range of values of $x,y,z$ to make the objective function bounded.