I am new to proximal functions and I am currently stuck with the following problem:
How can I derive the proximal operator for this function: trace of the Hadamard product of an exponential matrix x
$$\text h(x)= tr(e^{x \circ\ x}) $$
The derivative of h(x):
$$\text h'(x)= (e^{x \circ\ x}) \circ\ 2x $$
Edit: I would like my proximal operator to be defined in terms of squared Euclidean distance (to a nonconvex closed set) $$\text{d}(x) = \inf_{y\in C} \left\|x-y\right\|_2 $$ $$\text f(x)= \frac {d(x)^2 }{2}$$
$$\text prox_{\lambda f} (x) = \frac {1 }{1+\lambda} x + \frac {\lambda }{1+\lambda} P_C(x) $$
I would appreciate some help to derive the proximal operator if it can be done analytically.
I have reviewed https://archive.siam.org/books/mo25/mo25_ch6.pdf and also http://proximity-operator.net/multivariatefunctions.html.
The prox is defined to be
$$\mbox{prox}_f(x) = \arg\min_{y}f(x) + \frac{1}{2}\|x-y\|^2$$