The help I need here is mostly conceptual, but I suspect that either JuMP.jl or Convex.jl could do this. Any suggestions or pointers to texts/tutorials would be appreciated, I just don’t know how to reformulate this.

I need to solve a system in n+1 unknowns s_0, s_1, \dots, s_n (the first one is handled specially), described as follows.

Let x^+ = \max(0, x) as usual, and apply elementwise for vectors. Let \mathbf{s} denote the vector [s_1, \dots, s_n] (note: s_0 is not in there).

Let a > 0, b > 0, c_0 > 0, \kappa > 0 be constants, and \mathbf{c} and \mathbf{f} n-element vectors. Then the system is

and for all i = 1, \dots, n,

where \langle \cdot, \cdot \rangle is the dot product.

I read a suggestion that such problems can be cast as LP problems, but I have never encountered this before so I don’t know how to start.