Quadratic Integral Penalty Methods for Numerical Trajectory Optimization

TL;DR

This thesis introduces quadratic integral penalty algorithms for solving dynamic optimization problems, emphasizing convergence, avoiding numerical artifacts, and demonstrating effectiveness through computational experiments.

Contribution

It proposes three novel methods for dynamic optimization, including a direct transcription with quadratic penalties, a modified augmented Lagrangian, and a combined penalty and barrier approach.

Findings

Methods effectively solve dynamic optimization problems.

Algorithms avoid numerical artifacts like ringing.

Convergence is rigorously proven and demonstrated numerically.

Abstract

This thesis presents new mathematical algorithms for the numerical solution of a mathematical problem class called \emph{dynamic optimization problems}. These are mathematical optimization problems, i.e., problems in which numbers are sought that minimize an expression subject to obeying equality and inequality constraints. Dynamic optimization problems are distinct from non-dynamic problems in that the sought numbers may vary over one independent variable. This independent variable can be thought of as, e.g., time. This thesis presents three methods, with emphasis on algorithms, convergence analysis, and computational demonstrations. The first method is a direct transcription method that is based on an integral quadratic penalty term. The purpose of this method is to avoid numerical artifacts such as ringing or erroneous/spurious solutions that may arise in direct collocation methods. The second method is a modified augmented Lagrangian method that leverages ideas from augmented Lagrangian methods for the solution of optimization problems with large quadratic penalty terms, such as they arise from the prior direct transcription method. Lastly, we present a direct transcription method with integral quadratic penalties and integral logarithmic barriers. All methods are motivated with applications and examples, analyzed with complete proofs for their convergence, and practically verified with numerical experiments.