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Using multiobjective optimization to map the entropy region

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  • László Csirmaz

Abstract

Mapping the structure of the entropy region of at least four jointly distributed random variables is an important open problem. Even partial knowledge about this region has far reaching consequences in other areas in mathematics, like information theory, cryptography, probability theory and combinatorics. Presently, the only known method of exploring the entropy region is, or equivalent to, the one of Zhang and Yeung from 1998. Using some non-trivial properties of the entropy function, their method is transformed to solving high dimensional linear multiobjective optimization problems. Benson’s outer approximation algorithm is a fundamental tool for solving such optimization problems. An improved version of Benson’s algorithm is presented, which requires solving one scalar linear program in each iteration rather than two or three as in previous versions. During the algorithm design, special care is taken for numerical stability. The implemented algorithm is used to verify previous statements about the entropy region, as well as to explore it further. Experimental results demonstrate the viability of the improved Benson’s algorithm for determining the extremal set of medium-sized numerically ill-posed optimization problems. With larger problem sizes, two limitations of Benson’s algorithm is observed: the inefficiency of the scalar LP solver, and the unexpectedly large number of intermediate vertices. Copyright Springer Science+Business Media New York 2016

Suggested Citation

  • László Csirmaz, 2016. "Using multiobjective optimization to map the entropy region," Computational Optimization and Applications, Springer, vol. 63(1), pages 45-67, January.
    • Handle: RePEc:spr:coopap:v:63:y:2016:i:1:p:45-67
    DOI: 10.1007/s10589-015-9760-6
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    References listed on IDEAS

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    1. Matthias Ehrgott & Andreas Löhne & Lizhen Shao, 2012. "A dual variant of Benson’s “outer approximation algorithm” for multiple objective linear programming," Journal of Global Optimization, Springer, vol. 52(4), pages 757-778, April.
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    Cited by:

    1. Andreas Löhne & Andrea Wagner, 2017. "Solving DC programs with a polyhedral component utilizing a multiple objective linear programming solver," Journal of Global Optimization, Springer, vol. 69(2), pages 369-385, October.
    2. Samira Fallah & Ted K. Ralphs & Natashia L. Boland, 2024. "On the relationship between the value function and the efficient frontier of a mixed integer linear optimization problem," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 100(1), pages 175-220, August.

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