# Separable Convex Optimization with Nested Lower and Upper Constraints

## TL;DR

This paper introduces an efficient, gradient-free divide-and-conquer algorithm for solving convex resource allocation problems with nested bounds, achieving state-of-the-art complexity and practical performance for large-scale instances.

## Contribution

The paper presents a novel divide-and-conquer algorithm that handles nested lower and upper constraints without requiring strict convexity or differentiability, improving computational efficiency.

## Key findings

- Algorithm solves large problems with up to 1,000,000 variables in seconds.
- Achieves optimal complexity bounds for the problem class.
- Demonstrates promising applications to support vector ordinal regression.

## Abstract

We study a convex resource allocation problem in which lower and upper bounds are imposed on partial sums of allocations. This model is linked to a large range of applications, including production planning, speed optimization, stratified sampling, support vector machines, portfolio management, and telecommunications. We propose an efficient gradient-free divide-and-conquer algorithm, which uses monotonicity arguments to generate valid bounds from the recursive calls, and eliminate linking constraints based on the information from sub-problems. This algorithm does not need strict convexity or differentiability. It produces an $\epsilon$-approximate solution for the continuous problem in $\mathcal{O}(n \log m \log \frac{n B}{\epsilon})$ time and an integer solution in $\mathcal{O}(n \log m \log B)$ time, where $n$ is the number of decision variables, $m$ is the number of constraints, and $B$ is the resource bound. A complexity of $\mathcal{O}(n \log m)$ is also achieved for the linear and quadratic cases. These are the best complexities known to date for this important problem class. Our experimental analyses confirm the good performance of the method, which produces optimal solutions for problems with up to 1,000,000 variables in a few seconds. Promising applications to the support vector ordinal regression problem are also investigated.