A Survey of Algorithms for Separable Convex Optimization with Linear Ascending Constraints

TL;DR

This paper surveys algorithms for solving separable convex optimization problems with linear ascending constraints, highlighting methods applicable to non-smooth and non-strictly convex functions, including a linear time solution for d-separable cases.

Contribution

It provides a comprehensive overview of current algorithms for this class of problems, including a new linear time algorithm for d-separable functions.

Findings

Algorithms applicable to non-smooth, non-strictly convex functions

Linear time algorithm for d-separable functions

Broad applicability to resource allocation problems

Abstract

The paper considers the minimization of a separable convex function subject to linear ascending constraints. The problem arises as the core optimization in several resource allocation scenarios, and is a special case of an optimization of a separable convex function over the bases of a polymatroid with a certain structure. The paper presents a survey of state-of-the-art algorithms that solve this optimization problem. The algorithms are applicable to the class of separable convex objective functions that need not be smooth or strictly convex. When the objective function is a so-called $d$-separable function, a simpler linear time algorithm solves the problem.