Algorithms for the continuous nonlinear resource allocation problem---new implementations and numerical studies

TL;DR

This paper updates the literature survey on continuous nonlinear resource allocation problems, introduces improved algorithms with better pegging strategies, and evaluates their performance on large-scale problems, finding the relaxation algorithm most effective.

Contribution

It extends previous surveys with new references, enhances existing algorithms with novel pegging improvements, and provides comprehensive numerical comparisons for large problem instances.

Findings

Relaxation algorithm with pegging improvements performs best.

Practical time complexity for algorithms is linear up to 30 million variables.

Modified algorithms outperform previous versions in efficiency.

Abstract

Patriksson (2008) provided a then up-to-date survey on the continuous,separable, differentiable and convex resource allocation problem with a single resource constraint. Since the publication of that paper the interest in the problem has grown: several new applications have arisen where the problem at hand constitutes a subproblem, and several new algorithms have been developed for its efficient solution. This paper therefore serves three purposes. First, it provides an up-to-date extension of the survey of the literature of the field, complementing the survey in Patriksson (2008) with more then 20 books and articles. Second, it contributes improvements of some of these algorithms, in particular with an improvement of the pegging (that is, variable fixing) process in the relaxation algorithm, and an improved means to evaluate subsolutions. Third, it numerically evaluates several relaxation (primal) and breakpoint (dual) algorithms, incorporating a variety of pegging strategies, as well as a quasi-Newton method. Our conclusion is that our modification of the relaxation algorithm performs the best. At least for problem sizes up to 30 million variables the practical time complexity for the breakpoint and relaxation algorithms is linear.