hi guys,
i am using the convex matlab toolbox and i have the following convex problem:

variable w complex
min norm(w)
s.t. aw==1
max(abs(bw))<=1

is the problem above a convex problem? i minimize a convex function(norm), subject to a convex equality constraint and an inequality constraint. i know that both abs and max are convex functions but i am not sure for the max(abs(…))? my problem works for some cases of a and b but doesn’t work for some other cases of those vectors. any idea?

Have you examined the solver output in the NaN cases? Have you tried multiple solvers (SeDuMi and SDPT3)? The problem is not feasible (has no feasible solutions) for certain values of a and b. For example, in one dimension, if a = 1 and b = 2, then the problem is infeasible.

I think whether you have feasible solutions depend on the values of a and b. max(abs(\cdot)) is a convex function by the rule: “The composition of two convex functions is convex”.

That’s not quite the rule: the composition of a convex increasing function and a convex function is convex. The monotonicity of the outer function is essential.