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 set containment of a projection of the feasible
region and the uncertainty set. 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 quadratic problem with absolute values
functions. 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,
removing the absolute values and solving the problem can be decided with
linearly many polynomial optimization problems.
Lastly, the effectivity of the methods is tested on a variety of small cyclic
networks. For instances where robust feasibility or infeasibility can be
decided, level~2 or level~3 of the Lasserre relaxation hierarchy is typically
sufficient.