mosek

Tranform a constraint in Mosek

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

enter image description here

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

enter image description here

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

enter image description here

equivalent to

enter image description here

Solution

  • For the sake of completeness I repeat my comment as an answer. You can write

    t_i >= log(1 + exp(b_i^Tx-c_i))

    using two exponential cones as in https://docs.mosek.com/modeling-cookbook/expo.html#softplus-function This is a very special case of a more general log-sum-exp, namely log(exp(0) + exp(b_i^Tx-c_i)).

    Then the constraint becomes

    sum t_i <= N \log(2)

    If you use Rmosek then you can find pretty much ready code in https://docs.mosek.com/latest/rmosek/case-studies-logistic.html#doc-case-studies-logistic