-

 

 

 




Optimization Online





 

Affinely Adjustable Robust Linear Complementarity Problems

Christian Biefel (christian.biefel***at***fau.de)
Frauke Liers (frauke.liers***at***fau.de)
Jan Rolfes (jan.rolfes***at***fau.de)
Martin Schmidt (martin.schmidt***at***uni-trier.de)

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. Our main results are valid for the case of an uncertain LCP vector. Here, 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, in addition, the certain LCP matrix is positive semidefinite, we prove polynomial-time solvability and uniqueness of robust solutions. 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.

Keywords: Linear Complementarity Problems, Adjustable Robustness, Robust Optimization, Existence, Uniqueness

Category 1: Complementarity and Variational Inequalities

Category 2: Robust Optimization

Category 3: Integer Programming ((Mixed) Integer Linear Programming )

Citation:

Download: [PDF]

Entry Submitted: 08/13/2020
Entry Accepted: 08/13/2020
Entry Last Modified: 04/28/2021

Modify/Update this entry


  Visitors Authors More about us Links
  Subscribe, Unsubscribe
Digest Archive
Search, Browse the Repository

 

Submit
Update
Policies
Coordinator's Board
Classification Scheme
Credits
Give us feedback
Optimization Journals, Sites, Societies
Mathematical Optimization Society