Some Applications of Polynomial Optimization in Operations Research and Real-Time Decision Making
This work addresses real-time decision-making challenges for operations research and transportation engineering communities, presenting incremental advancements by adapting existing polynomial optimization methods to new applications.
The paper tackles real-time optimization and control problems in operations research and transportation engineering, such as wireless coverage, collision avoidance, and quadrotor hovering, by applying sum of squares (SOS) relaxations and SDSOS optimization techniques, achieving solutions for smaller-scale and real-time applications with semidefinite and second order cone programs.
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.