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

Aßmann D, Liers F, Stingl M, Vera JC (2018)

Publication Language: English

Publication Type: Journal article

Publication year: 2018

Journal

SIAM Journal on Optimization Society for Industrial and Applied Mathematics

Book Volume: 28

Pages Range: 2489-2517

Journal Issue: 3

DOI: 10.1137/17M112470X

Abstract

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 (SOS) 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.

Authors with CRIS profile

Denis Aßmann Sonderforschungsbereich/Transregio 154: Mathematische Modellierung, Simulation und Optimierung am Beispiel von Gasnetzwerken Frauke Liers-Bergmann Professur für Optimization under Uncertainty & Data Analysis Michael Stingl Lehrstuhl für Angewandte Mathematik (Kontinuierliche Optimierung)

Involved external institutions

Tilburg University NL Netherlands (NL)

How to cite

APA:

Aßmann, D., Liers, F., Stingl, M., & Vera, J.C. (2018). Deciding Robust Feasibility and Infeasibility Using a Set Containment Approach: An Application to Stationary Passive Gas Network Operations. SIAM Journal on Optimization, 28(3), 2489-2517. https://doi.org/10.1137/17M112470X

MLA:

Aßmann, Denis, et al. "Deciding Robust Feasibility and Infeasibility Using a Set Containment Approach: An Application to Stationary Passive Gas Network Operations." SIAM Journal on Optimization 28.3 (2018): 2489-2517.

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