SECOND-ORDER CONE OPTIMIZATION WITH A SINGLE SECOND-ORDER CONE

We consider the standard conic optimization problem involving only
a single second-order cone and do not assume any type of
constraint qualification (e.g. Slater) condition. We first tackle
the associated feasibility problem, which amounts to deciding
whether the problem is strictly feasible, weakly feasible, weakly
infeasible or strongly infeasible. We show that this can be
determined without any iterative process by solving a few linear
equality systems. We then outline how this procedure can be
generalized to solve the original second-order cone optimization
problem. In particular, we obtain a proof of the fact that such
second-order cone problems can never exhibit a duality gap. We
also discuss the algorithmic complexity viewpoint and possible
generalizations of this procedure.