Journal Article

Linear complementarity representation of piecewise linear functions

Tomoaki Sakurai and Toshiaki Murofushi

in IMA Journal of Applied Mathematics

Volume 80, issue 4, pages 1178-1198
Published in print August 2015 | ISSN: 0272-4960
Published online October 2014 | e-ISSN: 1464-3634 | DOI: https://dx.doi.org/10.1093/imamat/hxu047
Linear complementarity representation of piecewise linear functions

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We investigate a representation of piecewise linear function containing a complementarity condition, and call it a linear complementarity representation. This representation is also known as ‘Bokh1’ or ‘state-variable representation’ in literature. We discuss two spacial types of representation, called P-representation and ULT-representation, and give an elementary proof to show that either one of these two types of representation completely characterizes any piecewise linear function. We can also demonstrate that a P-representation can be transformed into a ULT-representation, and show how it can be implemented in an algorithmic manner. Furthermore, as an application of linear complementarity representation, we will present a formula of linear complementarity representation of the Choquet integral model over a finite set, which is a model used in the area of decision-making as an aggregation function.

Keywords: Piecewise linear function; Linear complementarityrepresentation; P-matrix; ULT-matrix

Journal Article.  9666 words.  Illustrated.

Subjects: Applied Mathematics

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