Mathematical Programming

, Volume 143, Issue 1–2, pp 339–356

MM algorithms for geometric and signomial programming

Full Length Paper Series A

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.

Keywords

Arithmetic-geometric mean inequality Geometric programming Global convergence MM algorithm Linearly constrained quadratic programming Parameter separation Penalty method Signomial programming 

Mathematics Subject Classification (2000)

90C25 26D07 

Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2012

Authors and Affiliations

  1. 1.Departments of Biomathematics, Human Genetics, and StatisticsUniversity of CaliforniaLos AngelesUSA
  2. 2.Department of StatisticsNorth Carolina State UniversityRaleighUSA

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