# Constraints

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(b
w))<=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?

It should be a valid CVX program. The argument requirements for max and abs are provided in the online help, and are satisfied for your problem.

You haven’t specified what you mean by “works” in “my problem works for some cases of a and b but doesn’t work for some other cases of those vectors”.

I’ll save the lecture on your not using CVX “correctly” for mcg, but please read FAQ: Why doesn’t CVX accept my problem? [READ THIS FIRST].

by ‘works’ i mean that it gives results in cases when it works, in other cases in gives NaN which means the solver was not able to find solutions.

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.

Thanks for correcting.

could you please elaborate the idea?