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In this paper we study feasibility and infeasibility of nonlinear two-stagefully adjustable robust feasibility problems with an empty first stage. This isequivalent to deciding whether the uncertainty set is contained within theprojection of the feasible region onto the uncertainty-space. Moreover, theconsidered sets are assumed to be described by polynomials. For answering thisquestion, two very general approaches using methods from polynomialoptimization are presented - one for showing feasibility and one for showinginfeasibility. The developed methods are approximated through sum of squarespolynomials and solved using semidefinite programs. Deciding robust feasibilityand infeasibility is important for gas network operations, which is a nonconvexfeasibility problem where the feasible set is described by a composition ofpolynomials with the absolute value function. Concerning the gas networkproblem, different topologies are considered. It is shown that a treestructured network can be decided exactly using linear programming.Furthermore, a method is presented to reduce a tree network with one additionalarc to a single cycle network. In this case, the problem can be decided byeliminating the absolute value functions and solving the resulting linearlymany polynomial optimization problems. Lastly, the effectivity of the methodsis tested on a variety of small cyclic networks. It turns out that forinstances where robust feasibility or infeasibility can be decidedsuccessfully, level 2 or level 3 of the Lasserre relaxation hierarchy typicallyis sufficient.

Deciding Robust Feasibility and Infeasibility Using a Set Containment Approach: An Application to Stationary Passive Gas Network Operations