MM Algorithms for Geometric and Signomial Programming
For optimization researchers, this provides a novel algorithmic framework for signomial programming, though the results are primarily theoretical with simple examples rather than broad empirical validation.
This paper develops new MM algorithms for signomial programming, reducing unconstrained problems to sequences of one-dimensional minimizations via the geometric-arithmetic mean and supporting hyperplane inequalities. The algorithms handle constraints and achieve linear convergence to interior points, with applications to constrained quadratic programming.
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.