  In this paper we study feasibility and infeasibility of nonlinear two-stage
fully adjustable robust feasibility problems with an empty first stage. This is
equivalent to deciding whether the uncertainty set is contained within the
projection of the feasible region onto the uncertainty-space. Moreover, the
considered sets are assumed to be described by polynomials. For answering this
question, two very general approaches using methods from polynomial
optimization are presented - one for showing feasibility and one for showing
infeasibility. The developed methods are approximated through sum of squares
polynomials and solved using semidefinite programs. Deciding robust feasibility
and infeasibility is important for gas network operations, which is a nonconvex
feasibility problem where the feasible set is described by a composition of
polynomials with the absolute value function. Concerning the gas network
problem, different topologies are considered. It is shown that a tree
structured network can be decided exactly using linear programming.
Furthermore, a method is presented to reduce a tree network with one additional
arc to a single cycle network. In this case, the problem can be decided by
eliminating the absolute value functions and solving the resulting linearly
many polynomial optimization problems. Lastly, the effectivity of the methods
is tested on a variety of small cyclic networks. It turns out that for
instances where robust feasibility or infeasibility can be decided
successfully, level 2 or level 3 of the Lasserre relaxation hierarchy typically
is sufficient.
