Numerical methods for accurate constraint implementation in optimization and optimal control
File(s)
Author(s)
Nita, Lucian
Type
Thesis
Abstract
This thesis examines different methods for discretizing dynamic optimization problems and numerically approximating the solution. The main challenge in problems that involve time dynamics is seeking a continuous-time solution with a limited number of decision variables. This is addressed using discrete-time approximations formulated as nonlinear programs (NLPs). In real-time control, the added challenge lies in finding feasible solutions quickly, as most NLP
solvers fail to return feasible points upon early termination. This research focuses on fast constraint satisfaction and incrementally improving objectives.
Two methods are presented and analyzed in this thesis. The first one employs an adaptive discretization for rapid non-smoothness detection, joining an integrated residual transcription with a flexible mesh where nodes are treated as decision variables. This method estimates errors during solve using numerical quadratures, improving accuracy for smooth problems while effectively capturing discontinuities. Flexible meshing reduces the number of nodes required to
compute a satisfactory solution and improves the memory use. Flexible mesh performs best when combined with integrated residual transcription, which, unlike collocation, accounts for inter-nodal errors. This approach prevents the clustering of nodes in regions with smoother dynamics or simpler solutions, ensuring a more uniform error distribution and improved solution accuracy.
The second method reformulates constrained optimization problems as a series of feasibility problems, each parameterized by an upper bound on the cost function value. Algorithmic variations are tested using CUTEst, and numerical experiments confirm the method’s proven convergence. Additionally, the results highlight properties such as monotonicity and convexity of the chosen residual function, which can be exploited to enhance the computational efficiency,
particularly for convex problems. The methods are presented in an applied manner, balancing theory and practical implementation.
Case studies and numerical experiments illustrate their effectiveness, as well as testing their limits on challenging problems.
solvers fail to return feasible points upon early termination. This research focuses on fast constraint satisfaction and incrementally improving objectives.
Two methods are presented and analyzed in this thesis. The first one employs an adaptive discretization for rapid non-smoothness detection, joining an integrated residual transcription with a flexible mesh where nodes are treated as decision variables. This method estimates errors during solve using numerical quadratures, improving accuracy for smooth problems while effectively capturing discontinuities. Flexible meshing reduces the number of nodes required to
compute a satisfactory solution and improves the memory use. Flexible mesh performs best when combined with integrated residual transcription, which, unlike collocation, accounts for inter-nodal errors. This approach prevents the clustering of nodes in regions with smoother dynamics or simpler solutions, ensuring a more uniform error distribution and improved solution accuracy.
The second method reformulates constrained optimization problems as a series of feasibility problems, each parameterized by an upper bound on the cost function value. Algorithmic variations are tested using CUTEst, and numerical experiments confirm the method’s proven convergence. Additionally, the results highlight properties such as monotonicity and convexity of the chosen residual function, which can be exploited to enhance the computational efficiency,
particularly for convex problems. The methods are presented in an applied manner, balancing theory and practical implementation.
Case studies and numerical experiments illustrate their effectiveness, as well as testing their limits on challenging problems.
Version
Open Access
Date Issued
2025-02-01
Date Awarded
01/12/2025
License URL
Advisor
Kerrigan, Eric
Sponsor
Engineering and Physical Sciences Research Council
Grant Number
EP/T51780X/1
Publisher Department
Department of Electrical and Electronic Engineering
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)