Some Applications of Polynomial Optimization in Operations Research and Real-Time Decision Making

TL;DR

This paper explores the use of algebraic polynomial optimization techniques, including SOS and SDSOS methods, for real-time decision making in operations research and transportation, demonstrating applications in wireless coverage, collision avoidance, and drone control.

Contribution

It introduces the first real-time applications of sum of squares techniques in optimization and control, using SDSOS optimization for larger-scale problems.

Findings

Successfully applied SOS and SDSOS methods to real-time problems

Demonstrated polynomial optimization in wireless coverage and UAV control

Achieved solutions with semidefinite and second order cone programming

Abstract

We demonstrate applications of algebraic techniques that optimize and certify polynomial inequalities to problems of interest in the operations research and transportation engineering communities. Three problems are considered: (i) wireless coverage of targeted geographical regions with guaranteed signal quality and minimum transmission power, (ii) computing real-time certificates of collision avoidance for a simple model of an unmanned vehicle (UV) navigating through a cluttered environment, and (iii) designing a nonlinear hovering controller for a quadrotor UV, which has recently been used for load transportation. On our smaller-scale applications, we apply the sum of squares (SOS) relaxation and solve the underlying problems with semidefinite programming. On the larger-scale or real-time applications, we use our recently introduced "SDSOS Optimization" techniques which result in second order cone programs. To the best of our knowledge, this is the first study of real-time applications of sum of squares techniques in optimization and control. No knowledge in dynamics and control is assumed from the reader.