Concise (short and yet sufficient for our purposes) introduction to (linear) complementarity problems and systems is in [1]. Besides describing the idea of complementarity in dynamical systems, it also shows how it is related to other modeling frameworks for hybrid dynamical systems. More detailed and yet very accessible introduction is in the thesis [2]. Condensed treatment is in the papers [3] and [4].
A readable introduction to the Extended Linear Complementarity Problem is in [5] (it is also freely available as a technical report).
The topics of complementarity constraints in dynamical systems and optimization is still being actively researched. A recent publication on QP optimization with complementarity constraints (LCQP) is [6].
Numerical methods for nonsmooth dynamical systems that are based on complementary constraints (and implemented in SICONOS software) are comprehensively presented in [7].
Back to topReferences
[1]
W. P. M. H. Heemels, B. De Schutter, and A. Bemporad,
“Equivalence of hybrid dynamical models,” Automatica, vol. 37, no. 7, pp. 1085–1091, Jul. 2001, doi:
10.1016/S0005-1098(01)00059-0.
[3]
W. P. M. H. Heemels, J. M. Schumacher, and S. Weiland,
“Linear Complementarity Systems,” SIAM Journal on Applied Mathematics, vol. 60, no. 4, pp. 1234–1269, Jan. 2000, doi:
10.1137/S0036139997325199.
[4]
A. J. van der Schaft and J. M. Schumacher,
“Complementarity modeling of hybrid systems,” IEEE Transactions on Automatic Control, vol. 43, no. 4, pp. 483–490, Apr. 1998, doi:
10.1109/9.664151.
[5]
B. De Schutter and B. De Moor,
“The Extended Linear Complementarity Problem and the Modeling and Analysis of Hybrid Systems,” in
Hybrid Systems V, P. Antsaklis, M. Lemmon, W. Kohn, A. Nerode, and S. Sastry, Eds., in Lecture
Notes in
Computer Science. Berlin, Heidelberg: Springer, 1999, pp. 70–85. doi:
10.1007/3-540-49163-5_4.
[6]
J. Hall, A. Nurkanovic, F. Messerer, and M. Diehl,
“LCQPow – A Solver for Linear Complementarity Quadratic Programs.” arXiv, Nov. 2022. Accessed: Dec. 03, 2022. [Online]. Available:
http://arxiv.org/abs/2211.16341
[7]
V. Acary and B. Brogliato,
Numerical Methods for Nonsmooth Dynamical Systems: Applications in Mechanics and Electronics. in Lecture
Notes in
Applied and
Computational Mechanics, no. 35. Berlin Heidelberg: Springer, 2008. Available:
https://doi.org/10.1007/978-3-540-75392-6