Convex separable problems with linear and box constraints in signal processing and communications

TL;DR

This paper presents a simple, closed-form solution for a class of convex optimization problems with linear and box constraints, applicable to power allocation in signal processing and communications.

Contribution

It introduces a novel, intuitive algorithm with graphical interpretation for solving separable convex problems with linear and box constraints.

Findings

Algorithm computes solutions in finite steps

Graphical interpretation provides intuitive understanding

Applicable to power allocation problems in communications

Abstract

In this work, we focus on separable convex optimization problems with box constraints and a set of triangular linear constraints. The solution is given in closed-form as a function of some Lagrange multipliers that can be computed through an iterative procedure in a finite number of steps. Graphical interpretations are given casting valuable insights into the proposed algorithm and allowing to retain some of the intuition spelled out by the water-filling policy. It turns out that it is not only general enough to compute the solution to different instances of the problem at hand but also remarkably simple in the way it operates. We also show how some power allocation problems in signal processing and communications can be solved with the proposed algorithm.