I want to optimize the following function:
$$ argmin_{x} f(x) = g(x) + h(x) $$
, where I can get $\nabla_xg(x)$, but cannot calculate $\nabla_xh(x)$.
The derivative-free method, such as the Hill climbing method, works well, but I wonder whether I can make use of $\nabla_xg(x)$.
Problems of this type may be solved with splitting methods. One very popular case if the proximal gradient method. If $h$ is convex and lower-semicontinuous and if you are able to caluculate the proximal map, namely $$\operatorname{prox}_{\lambda h}(x) = \operatorname{argmin}_y \tfrac12\|y-x\|_2^2 + \lambda h(y)$$ you can iterate $$x^{k+1} = \operatorname{prox}_{\lambda h}(x^k - \lambda\nabla g(x^k)).$$ If $g$ is convex and $\nabla g$ is Lipschitz continuous with constant $L$ this iteration will converge to a solution if $0<\lambda < 2/L$. There are extensions for non-convex problems as well…
