Subgradient Algorithm, Stochastic Subgradient Algorithm, Incremental Subgradient Algorithm, and Set Location Problems

TL;DR

This paper introduces simple subgradient algorithms for solving new facility location models involving sets, generalizing classical problems, and demonstrates their broad applicability and ease of implementation in convex optimization contexts.

Contribution

The paper develops and presents easy-to-implement subgradient algorithms for new set-based facility location models, extending classical problems like Fermat-Torricelli.

Findings

Algorithms are simple and easy to implement.

Applicable to a broad range of convex optimization problems.

Suitable for teaching and learning convex optimization.

Abstract

In recent years, important progress has been made in applying methods and techniques of convex optimization to many fields of applications such as location science, engineering, computational statistics, and computer science. In this paper, we present some simple algorithms for solving a number of new models of facility location involving sets which generalize the classical Fermat-Torricelli problem and the smallest enclosing circle problem. The general nondifferentiability of the models prevents us from applying gradient-type algorithms, so our approach is to use subgradient-type algorithms instead. The algorithms presented in this paper are easy to implement and applicable for a broad range of problems that are also suitable for teaching and learning convex optimization.