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

  • 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.

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Author:Denis Aßmann, Frauke Liers, Michael Stingl, Juan Vera
Document Type:Preprint
Date of Publication (online):2017/03/19
Date of first Publication:2017/03/19
Release Date:2017/03/19
Tag:polynomial optimization; robust optimization; stationary gas transport
Institutes:Friedrich-Alexander-Universität Erlangen-Nürnberg
Tilburg University