Affinely Adjustable Robust Linear Complementarity Problems
Biefel C, Liers F, Rolfes J, Schmidt M (2022)
Publication Language: English
Publication Status: Submitted
Publication Type: Journal article, Original article
Future Publication Type: Other publication type
Publication year: 2022
Journal
Book Volume: 32
Pages Range: 152-172
Journal Issue: 1
DOI: 10.1137/20M1359778
Open Access Link: https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/319
Abstract
Linear complementarity problems are a powerful tool for modeling many practically relevant situations such as market equilibria. They also connect many sub-areas of mathematics like game theory, optimization, and matrix theory. Despite their close relation to optimization, the protection of LCPs against uncertainties - especially in the sense of robust optimization - is still in its infancy. During the last years, robust LCPs have only been studied using the notions of strict and Γ-robustness. Unfortunately, both concepts lead to the problem that the existence of robust solutions cannot be guaranteed. In this paper, we consider affinely adjustable robust LCPs. In the latter, a part of the LCP solution is allowed to adjust via a function that is affine in the uncertainty. We show that this notion of robustness allows to establish strong characterizations of solutions for the cases of uncertain matrix and vector, separately, from which existence results can be derived. For an uncertain LCP vector, we additionally provide sufficient conditions on the LCP matrix for the uniqueness of a solution. Moreover, based on characterizations of the affinely adjustable robust solutions, we derive a mixed-integer programming formulation that allows to solve the corresponding robust counterpart. If the LCP matrix is uncertain, characterizations of solutions are developed for every nominal matrix, i.e., these characterizations are, in particular, independent of the definiteness of the nominal matrix. Robust solutions are also shown to be unique for positive definite LCP matrix but both uniqueness and mixed-integer programming formulations still remain open problems if the nominal LCP matrix is not positive definite.
Authors with CRIS profile
Christian Biefel
Professur für Optimization under Uncertainty & Data Analysis
Frauke Liers-Bergmann
Professur für Optimization under Uncertainty & Data Analysis
Jan Rolfes
Lehrstuhl für Analytics & Mixed-Integer Optimization
Related research project(s)
Robustification of Physics Parameters in Gas Networks (B06) (2018 - 2022)
TRR 154: Mathematical Modelling, Simulation and Optimisation Using the Example of Gas Networks (TRR 154)
July 1, 2018 - June 30, 2022
Decomposition methods for mixed-integer optimal control (A05) (2018 - 2022)
TRR 154: Mathematical Modelling, Simulation and Optimisation Using the Example of Gas Networks (TRR 154)
July 1, 2018 - June 30, 2022
Multilevel mixed-integer nonlinear optimization for gas markets (B08) (2018 - 2022)
TRR 154: Mathematical Modelling, Simulation and Optimisation Using the Example of Gas Networks (TRR 154)
July 1, 2018 - June 30, 2022
Energiemarktdesign (EMD)
Energie Campus Nürnberg (EnCN2)
Jan. 1, 2017 - Dec. 31, 2021
Involved external institutions
Germany (DE)How to cite
APA:
Biefel, C., Liers, F., Rolfes, J., & Schmidt, M. (2022). Affinely Adjustable Robust Linear Complementarity Problems. SIAM Journal on Optimization, 32(1), 152-172. https://dx.doi.org/10.1137/20M1359778
MLA:
Biefel, Christian, et al. "Affinely Adjustable Robust Linear Complementarity Problems." SIAM Journal on Optimization 32.1 (2022): 152-172.
BibTeX: Download