Proximal Operator for $ f \left( \boldsymbol{x} \right) = - \log \left(x_1 \cdot x_2 \cdot \ldots \cdot x_n \right) $

Question

What is the proximal operator for $f(\mathbf{x})=-\log(x_1x_2\dots x_n)$, $\mathbf{x}>0$ and its optimal solution?

It is easy to find the former,

$P_{f}(x)=\arg \min_{\mathbf{y}>0}-\log(y_1y_2\dots y_n)+\frac{1}{2}\|\mathbf{y}-\mathbf{x}\|^2_2,$

but solving it seems to be a different ball game. Can I get some help on this?

Answer 1

Your problem is separable in the coordinates:

$$−\log \Pi_{i=1}^n y_i+\frac{1}{2}\|y−x\|_2^2=\sum_{i=1}^n−\log y_i+\frac{1}{2}(y_i−x_i)^2. $$

So you can consider the one-dimensional case (i.e $n=1$), which has an algebraic solution obtainable trivially by applying the constructs detailed here to $1 \times 1$ positive-definite matrices (i.e positive scalars!).