Some applications of polynomial optimization in operations research and real-time decision making

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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: (1) wireless coverage of targeted geographical regions with guaranteed signal quality and minimum transmission power, (2) computing real-time certificates of collision avoidance for a simple model of an unmanned vehicle (UV) navigating through a cluttered environment, and (3) 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.

Original languageEnglish (US)
Pages (from-to)709-729
Number of pages21
JournalOptimization Letters
Volume10
Issue number4
DOIs
StatePublished - Apr 1 2016

All Science Journal Classification (ASJC) codes

  • Control and Optimization

Keywords

  • Algebraic techniques in optimization
  • Lyapunov methods
  • Polynomial optimization
  • Real-time optimization
  • Second order cone programming
  • Semidefinite programming

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