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MM algorithms for geometric and signomial programming

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Abstract

This paper derives new algorithms for signomial programming, a generalization of geometric programming. The algorithms are based on a generic principle for optimization called the MM algorithm. In this setting, one can apply the geometric-arithmetic mean inequality and a supporting hyperplane inequality to create a surrogate function with parameters separated. Thus, unconstrained signomial programming reduces to a sequence of one-dimensional minimization problems. Simple examples demonstrate that the MM algorithm derived can converge to a boundary point or to one point of a continuum of minimum points. Conditions under which the minimum point is unique or occurs in the interior of parameter space are proved for geometric programming. Convergence to an interior point occurs at a linear rate. Finally, the MM framework easily accommodates equality and inequality constraints of signomial type. For the most important special case, constrained quadratic programming, the MM algorithm involves very simple updates.

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Authors and Affiliations

  1. Departments of Biomathematics, Human Genetics, and Statistics, University of California, Los Angeles, CA, 90095-1766, USA

    Kenneth Lange

  2. Department of Statistics, North Carolina State University, 2311 Stinson Drive, Campus Box 8203, Raleigh, NC, 27695-8203, USA

    Hua Zhou

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  1. Kenneth Lange
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  2. Hua Zhou
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Correspondence to Hua Zhou.

Additional information

Research was supported by United States Public Health Service grants GM53275, MH59490, and R01HG006139.

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Lange, K., Zhou, H. MM algorithms for geometric and signomial programming. Math. Program. 143, 339–356 (2014). https://doi.org/10.1007/s10107-012-0612-1

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  • DOI: https://doi.org/10.1007/s10107-012-0612-1

Keywords

Mathematics Subject Classification (2000)