Refining and extending works by Ye and Kitahara-Mizuno, this paper presents new results on the number of pivots of simplex-type methods for solving linear programs of the Leontief kind, certain linear complementarity problems of the P kind, and nonnegative constrained convex quadratic programs. Our results contribute to the further understanding of the complexity and efficiency … Read more
We consider adjustable robust linear complementarity problems and extend the results of Biefel et al.~(2022) towards convex and compact uncertainty sets. Moreover, for the case of polyhedral uncertainty sets, we prove that computing an adjustable robust solution of a given linear complementarity problem is equivalent to solving a properly chosen mixed-integer linear feasibility problem. ArticleDownload … Read more
Linear complementarity problems (LCPs) are an important modeling tool for many practically relevant situations but also have many important applications in mathematics itself. Although the continuous version of the problem is extremely well studied, much less is known about mixed-integer LCPs (MILCPs) in which some variables have to be integer-valued in a solution. In particular, … Read more
We consider a novel class of linear bilevel optimization models with a lower level that is a linear program with complementarity constraints (LPCC). We present different single-level reformulations depending on whether the linear complementarity problem (LCP) as part of the lower-level constraint set depends on the upper-level decisions or not as well as on whether … Read more
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 … Read more
We study uncertain linear complementarity problems (LCPs), i.e., problems in which the LCP vector q or the LCP matrix M may contain uncertain parameters. To this end, we use the concept of Gamma-robust optimization applied to the gap function formulation of the LCP. Thus, this work builds upon [16]. There, we studied Gamma-robustified LCPs for … Read more
Complementarity problems are often used to compute equilibria made up of specifically coordinated solutions of different optimization problems. Specific examples are game-theoretic settings like the bimatrix game or energy market models like for electricity or natural gas. While optimization under uncertainties is rather well-developed, the field of equilibrium models represented by complementarity problems under uncertainty … Read more
The paper “An algorithmic characterization of P-matricity” (SIAM Journal on Matrix Analysis and Applications, 34:3 (2013) 904–916, by the same authors as here) implicitly assumes that the iterates generated by the Newton-min algorithm for solving a linear complementarity problem of dimension n, which reads 0 ⩽ x ⊥ (M x + q) ⩾ 0, are … Read more
This paper mainly studies two topics: linear complementarity problems for modeling electricity market equilibria and optimization under uncertainty. We consider both perfectly competitive and Nash–Cournot models of electricity markets and study their robustifications using strict robustness and the Γ-approach. For three out of the four combinations of economic competition and robustification, we derive algorithmically tractable … Read more
The plain Newton-min algorithm for solving the linear complementarity problem (LCP) 0 ≤ x ⊥ (Mx+q) ≥ 0 can be viewed as an instance of the plain semismooth Newton method on the equational version min(x,Mx+q) = 0 of the problem. This algorithm converges for any q when M is an M-matrix, but not when it … Read more