2023/06/13 by Willem Esterhuizen, Esterhuizen, Willem, Kathrin Flaßkamp +5
Computer Science · Mathematics · #49K15 #49M99 #90C30 #93B40 #93C55 #Advanced Optimization Algorithms Research #FOS: Electrical engineering #FOS: Mathematics #Optimization and Control (math.OC) #Polynomial and algebraic computation #Systems and Control (eess.SY) #electronic engineering #information engineering
paper · pdf · doi:10.48550/arxiv.2306.07852
openalex publication_date 2023/06/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Homotopy methods are attractive due to their capability of solving difficult optimisation and optimal control problems. The underlying idea is to construct a homotopy, which may be considered as a continuous (zero) curve between the difficult original problem and a related, comparatively easy one. Then, the solution of the easier one is continuously perturbed along the zero curve towards the sought-after solution of the original problem. We propose a methodology for the systematic construction of such zero curves for discrete-time optimal control problems drawing upon the theory of globally convergent homotopies for nonlinear programs. The proposed framework ensures that for almost every initial guess at a solution there exists a suitable homotopy path that is, in addition, numerically convenient to track. We demonstrate the results by solving optimal path planning problems for a linear system and the nonlinear nonholonomic car (Dubins' vehicle).