Find the extrema of $\sum_{i=1}^n u_i v_i \log \left| \frac{v_i}{u_i} \right|$

Question

This question is similar to the following one: Maximizing and minimizing dot products. However there are significant differences, hence I opened a new question.

Maximize and minimize

$$\sum_{i=1}^n u_i v_i \log \left| \frac{v_i}{u_i} \right|$$

such that $u,v \in \mathbb{R^n}$ with $\|u\| = 1$ ,$u_i > 0, v_i \geq -c_2$ where $c_2$ is a small positive number, and $\sum_{i=1}^n v_i= c$ where $c<1$.

The result in my opinion is $-c \log c$ for the maximum and $c \log c$ for the minimum. I tried solving it with Lagrange multipliers, but it doesn't work out, I'll be glad for help.

Answer 1

This is an answer to the problem with the constraint $v_i \ge 0$ and $n>1$.

Let $g(t) = t\log(t)$ for $t>0$ and $g(0)=0$, which is continuous on $[0,\infty)$ and strictly convex.

We can rephrase the problem with $v_i = u_i t_i$ and obtain the equivalent problem $$ f(u,t) = \sum_i u_i^2 g(t_i) $$ subject to $\|u\| = 1$, $u_i > 0$, $\sum_i u_i t_i = c$, $t_i \ge 0$, where $c\in(0,1)$ is fixed. Now, we can obtain following maximizer analytically:

Fix a $u\in\mathbb R^n$ with $\|u\| = 1$ and $0 < u_1 \le \dotsb u_n \le 1$. Let $f_u(t) = f(u,t) = \sum_i u_i^2 g(t_i)$.

1. As $g$ is continuous and convex, so is $f_u$ continuous convex on the compact and convex set $S_u = \{ t\in\mathrm R^n \mid t_i \ge 0, \sum_i u_i t_i = c \}$. Thus, the global maximum of $f_u$ is attained by an extreme point of $S_u$. That is, $$ \sup_{t\in S_u} f_u(t) = \max_{1\le i \le n} f_u\left( \frac c {u_i} e_i \right) = \max_{1\le i \le n} u_i^2 g\left( \frac c {u_i} \right).$$

2. Now consider $h(s) = s^2 g(c/s) = sc \log(c/s)$ for $s>0$ and $h(0) = 0$. We have $h'(s) = 0$ if and only if $s = c/e$, that is $$ \max_{0\le s \le 1} h(s) = \max\{ h(0), h(c/e), h(1) \} = \max\left\{0, \frac{c^2}{e}, c\log(c) \right\} = \frac{c^2}{e}. $$

3. That is, we have $$ \sup f = \sup_{u} \sup_{t\in S_u} f_u(t) \le \frac{c^2}{e},$$ which is attainable in case of $n>1$ by some $(u, t)$ with $u_n = c/e$, $t_n = e$, and $t_1=\dotsb = t_{n-1} = 0$.

Note: With the constraints $v_i \ge -c2$, the objective function won't be convex in $t_i = v_i/u_i$.