Tranform a constraint in Mosek

Question

I have to solve the following optimisation problem in R Mosek:

This is a convex constraint which can be transformed into the intersection of 2^N-1 cone constraints and one half space:

This is unfeasible in my actual case because N=50. What can I do? Is it my problem impossible to be solved (with R Mosek)?

---

Edit following the answer:

Is it my constraint

equivalent to

Answer 1

Update: See the comments first.

It is a log sum exp constraint that can easily be dealt with. Indeed your first constraint is equivalent to

\begin{array}{rcl}
\log{\sum_{i=0}^n e^{t_i}} & \leq & s, \\
t_0                        & =    & 0, \\
b_i^Tx-c_i - t_i           & =    & 0, \\
s                          & =    & n\log{2}. \\
\end{array}

PS. I could not get the math formatting to work.