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

arXiv: https://arxiv.org/abs/1504.06002 · HTML: https://mlsanity.com/paper/1504.06002/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Apr 2015.

## Built on by (most cited first, 11 of 11)

- [AutoCost: Evolving Intrinsic Cost for Zero-violation Reinforcement Learning](https://mlsanity.com/paper/2301.10339/) (2023)
- [Actor-Critic Physics-informed Neural Lyapunov Control](https://mlsanity.com/paper/2403.08448/) (2024)
- [Learning for Spatial Branching: An Algorithm Selection Approach](https://mlsanity.com/paper/2204.10834/) (2022)
- [Learning Conservative Neural Control Barrier Functions from Offline Data](https://mlsanity.com/paper/2505.00908/) (2025)
- [Neural Control and Certificate Repair via Runtime Monitoring](https://mlsanity.com/paper/2412.12996/) (2024)
- [Gaussian Control Barrier Functions : A Non-Parametric Paradigm to Safety](https://mlsanity.com/paper/2203.15474/) (2022)
- [Learning Polynomial Problems with SL(2,ℝ) Equivariance](https://mlsanity.com/paper/2312.02146/) (2023)
- [Certified Policy Verification and Synthesis for MDPs under Distributional Reach-avoidance Properties](https://mlsanity.com/paper/2405.04015/) (2024)
- [Combining Neural Networks and Symbolic Regression for Analytical Lyapunov Function Discovery](https://mlsanity.com/paper/2406.15675/) (2024)
- [Analytical Lyapunov Function Discovery: An RL-based Generative Approach](https://mlsanity.com/paper/2502.02014/) (2025)
- [Towards Solving Polynomial-Objective Integer Programming with Hypergraph Neural Networks](https://mlsanity.com/paper/2603.19318/) (2026)

## Abstract (arXiv)

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.