MM Algorithms for Geometric and Signomial Programming

TL;DR

This paper introduces new MM algorithms for signomial programming, extending geometric programming techniques with convergence guarantees and the ability to handle constraints, enabling efficient optimization in complex scenarios.

Contribution

The paper develops a novel MM-based framework for signomial programming, including convergence analysis and simple update rules for constrained quadratic cases.

Findings

Algorithms can converge to boundary or interior points.

Conditions for uniqueness and interior convergence are established.

Linear convergence rate for interior solutions.

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.