Separable convex optimization problems with linear ascending constraints

TL;DR

This paper presents a finite-step algorithm for solving separable convex optimization problems with linear ascending constraints, demonstrating monotonicity and convexity properties of the optimal value relative to constraint parameters.

Contribution

It introduces a novel algorithm for these problems under an ordering condition, with proven monotonicity and convexity of the optimal value.

Findings

Algorithm determines the optimum in finite steps

Optimal value is monotone with respect to constraint parameters

Optimal value is convex with respect to parameters

Abstract

Separable convex optimization problems with linear ascending inequality and equality constraints are addressed in this paper. Under an ordering condition on the slopes of the functions at the origin, an algorithm that determines the optimum point in a finite number of steps is described. The optimum value is shown to be monotone with respect to a partial order on the constraint parameters. Moreover, the optimum value is convex with respect to these parameters. Examples motivated by optimizations for communication systems are used to illustrate the algorithm.