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Handbook of Combinatorial Optimization pp 3015–3056Cite as

Simplicial Methods for Approximating Fixed Point with Applications in Combinatorial Optimizations

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Abstract

Simplicial methods were originated by Scarf for approximating fixed points of continuous mappings. They have many applications in economics and science. With regard to its geometric structure, a simplicial method can be classified as either a variable dimension simplicial method or a homotopy simplicial method. A variable dimension simplicial method works directly on the interested space, whereas a simplicial homotopy method needs to introduce an extra dimension. It is well known that integer programming is equivalent to determining whether there is an integer point in a polytope. Simplicial methods were extended to computing an integer point in a polytope. There is a significant difference between simplicial methods for approximating fixed points and simplicial methods for integer programming, though they both have the same foundation. Two most important components of simplicial methods are labeling rules and triangulations. Efficiency of simplicial methods depends critically on the underlying triangulations. Three simplest triangulations of R n are the K 1-triangulation, the J 1-triangulation, and the D 1-triangulation. This chapter presents a brief introduction to these developments of simplicial methods.

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Acknowledgements

This work was partially supported by GRF: CityU 112809 of the Government of Hong Kong SAR.

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  1. Department of Systems Engineering and Engineering Management, City University of Hong Kong, Kowloon, Hong Kong

    Chuangyin Dang

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  1. Department of Industrial and Systems Eng, University of Florida, Gainesville, Florida, USA

    Panos M. Pardalos

  2. Department of Computer Science, University of Texas, Dallas, Richardson, Texas, USA

    Ding-Zhu Du

  3. Dept. Comp. Sci. & Engineering, University of California, San Diego, La Jolla, California, USA

    Ronald L. Graham

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Dang, C. (2013). Simplicial Methods for Approximating Fixed Point with Applications in Combinatorial Optimizations. In: Pardalos, P., Du, DZ., Graham, R. (eds) Handbook of Combinatorial Optimization. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-7997-1_79

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