**Reference:**

B. De Schutter,
J. Schepers, and
I. Van Mechelen,
"On algorithms for a binary-real (max,×) matrix approximation
problem," *Proceedings of the 45th IEEE Conference on Decision and
Control*, San Diego, California, pp. 5168-5173, Dec. 2006.

**Abstract:**

We consider algorithms to solve the problem of approximating a given
matrix D with the (max,×) product of a binary (i.e., a 0-1)
matrix S and a real matrix P: min_{S,P} || S · P - D
||. The norm to be used is the l_{1}, l_{2} or
l_{∞} norm, and the (max,×) matrix product is
constructed in the same way as the conventional matrix product, but
with addition replaced by maximization. This approximation problem
arises among others in data clustering applications where the maximal
component instead of the sum of the components determines the final
result. We propose several algorithms to address this problem. The
binary-real (max,×) matrix approximation problem can be solved
exactly using mixed-integer programming, but since this approach
suffers from combinatorial explosion we also propose some alternative
approaches based on alternating nonlinear optimization, and a method
to obtain good initial solutions. We conclude with a simulation study
in which the performance and optimality of the different algorithms
are compared.

Corresponding technical report: pdf file (136 KB)

@inproceedings{DeSSch:06-027,

author={B. {D}e Schutter and J. Schepers and I. {V}an Mechelen},

title={On algorithms for a binary-real $(\max,\times)$ matrix approximation problem},

booktitle={Proceedings of the 45th IEEE Conference on Decision and Control},

address={San Diego, California},

pages={5168--5173},

month=dec,

year={2006}

}

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