Transform an optimisation problem for MOSEK

Question

I would like to use Mosek to solve the following problem:

The constraint is convex. In the guidance of the problems that Mosek can solve I could not find a "close" example. Hence, I wonder: (1) Is Mosek suitable to solve the problem above? (2) If yes, how can I readapt the problem above to be solved by Mosek? (3) If not, could you suggest an alternative solver I might use?

Answer 1

Yes, the upper bound on softplus function, or more general log-sum-exp, can be modeled with the exponential cone like here https://docs.mosek.com/modeling-cookbook/expo.html#softplus-function

Here is an example where log-sum-exp is used in a bigger problem https://docs.mosek.com/latest/pythonfusion/case-studies-logistic.html#doc-case-studies-logistic

Many modeling tools that can use Mosek as a solver will have a log_sum_exp atom available directly, for instance see https://www.cvxpy.org/tutorial/functions/index.html