How to find maximum without local maxima

Question

I want to maximize a function that has no local maximum in a certain interval. For instance, consider the function $f(x)=x$ over the interval $[0,1]$. There is no local maximum, but the maximum value of the function is clearly $1$. Here is the function that I would like to maximize: $$f(x_1,x_2,...,x_n)=\frac{1}{\Big(1-\sum\limits_{k=1}^{n}x_k\Big)^{1-\sum\limits_{k=1}^{n}x_k}\prod\limits_{k=1}^{n}x_k^{x_k}}$$

For instance, if $k=3$, we have $$f(x_1,x_2,x_3)=\frac{1}{(1-x_1-x_2-x_3)^{1-x_1-x_2-x_3}x_1^{x_1}x_2^{x_2}x_3^{x_3}}$$

I want to maximize this function with the following constraints: $$\sum\limits_{k=1}^{n}kx_k\leq 1$$ $$0\leq x_i\leq 1$$ This family of functions seems to have no local maximima when the constraints are applied. How can I work around this and find a maximal value(s)?

Answer 1

Have you tried using NMaximize? For instance:

f[x1_, x2_, x3_] := 1/(Abs[1-x1-x2-x3]^(1-x1-x2-x3) x1^x1 x2^x2 x3^x3)

NMaximize[
    {
    f[x1, x2, x3],
    0<x1<1 && 0<x2<1 && 0<x3<1 && x1 + 2 x2 + 3 x3 < 1
    },
    {x1, x2, x3}
]

{3.61072, {x1 -> 0.276953, x2 -> 0.182041, x3 -> 0.119655}}