I am stuck to the problem 4 here, course Mat-2.3139, the due day was yesterday. The hint is "Optimality-condition for a convex-problem". I have asked this now from 3 assistants and everyone with slighlty different answers: one assistant would assume $L^\infty$ -smoothness immediately but the assignment does not specify anything about smoothness and other two assistant got somehow lost. Wikipedia here states:
"The necessary conditions are sufficient for optimality if the objective function and the inequality constraints are continuously differentiable convex functions and the equality constraints are affine functions."
but the problem does not specify the $L^\infty$ so I cannot use KKT. So what is the condition the assignment is looking for here? Source is here. [Update] The original statement did have the bug with
"the lack of differentiability", so Lipschitz at least according to Willie Wong, so no necessary conditions found. So let's consider the corrected version. I think this was the purpose of the assignment, I wrote it in my own words below.