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MM Algorithms for Geometric and Signomial Programming.

Kenneth Lange1, Hua Zhou2

  • 1Departments of Biomathematics, Human Genetics, and Statistics, University of California, Los Angeles, CA 90095-1766, USA. klange@ucla.edu.

Mathematical Programming
|March 18, 2014
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Summary
This summary is machine-generated.

New algorithms for signomial programming, a generalization of geometric programming, are derived using the MM algorithm. These methods simplify complex optimization problems into a series of one-dimensional minimizations, offering efficient solutions for constrained quadratic programming.

Keywords:
MM algorithmarithmetic-geometric mean inequalitygeometric programmingglobal convergencelinearly constrained quadratic program-mingparameter separationpenalty methodsignomial programming

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Area of Science:

  • Optimization Theory
  • Mathematical Programming

Background:

  • Signomial programming is a generalization of geometric programming, presenting complex optimization challenges.
  • Existing methods may struggle with the complexity of signomial functions and constraints.

Purpose of the Study:

  • To develop novel algorithms for solving signomial programming problems.
  • To leverage the MM algorithm for efficient optimization.
  • To extend the applicability of these methods to constrained problems.

Main Methods:

  • Application of the geometric-arithmetic mean inequality.
  • Utilization of supporting hyperplane inequalities to create surrogate functions.
  • Reduction of unconstrained signomial programming to one-dimensional minimization.
  • Adaptation of the MM framework for equality and inequality constraints.

Main Results:

  • The MM algorithm effectively reduces signomial programming to sequential one-dimensional minimizations.
  • Demonstrated convergence to boundary points or continua of minimum points.
  • Established conditions for unique interior minimum points in geometric programming.
  • Achieved linear convergence rates for interior points.
  • Showcased simple updates for constrained quadratic programming.

Conclusions:

  • The MM algorithm provides a powerful and versatile framework for signomial programming.
  • The derived methods offer efficient solutions, particularly for constrained quadratic programming.
  • The approach simplifies complex optimization tasks through parameter separation and iterative refinement.