Modeling and Algorithms for Nonconvex TrajectoryGeneration Problems: From Constraint Reformulations and First-Order Proximal Methods to Structure-Exploiting Convex Solvers

Abstract

Trajectory generation plays a central role in modern Guidance, Navigation, and Control (GNC) systems by converting high-level mission objectives into reference trajectories that comply with system dynamics and task constraints. Among various methods, optimization-based formulation is favored for its ability to integrate dynamics, constraints, and performance objectives within a unified framework. Although various optimization-based trajectory generation methods have been successfully deployed in practice, many suffer from fundamental limitations that undermine their reliability. In particular, existing approaches often lack rigorous theoretical guarantees for convergence and constraint satisfaction, especially when implemented in discretized form. Furthermore, real-time deployment is frequently hindered by high computational costs and a heavy reliance on manual parameter tuning. To address these challenges, this dissertation presents three core contributions that improve the theoretical reliability and computational efficiency for trajectory generation. The first part of the dissertation focuses on a structured form of nonconvexity arising from thrust lower bounds, a setting where a method known as Lossless Convexification (LCvx) has been widely adopted in applications due to its empirical effectiveness. LCvx addresses this challenge by relaxing the original nonconvex problem as a convex optimization problem whose solution, under certain conditions, is also optimal for the nonconvex formulation. However, existing LCvx theory provides guarantees only for continuous-time optimal control problems, where trajectories and controls are modeled as functions over a continuous time domain. In contrast, practical implementations rely on discrete-time formulations, where the optimization variables are finite-dimensional vectors defined over temporal grids. This gap raises concerns about the theoretical soundness of applying LCvx in practical applications. In this dissertation, we extend LCvx theory to the discrete-time setting and establish formal guarantees that support its use in realistic implementations. In particular, we show that the solution to the convex problem resulting from LCvx satisfies the original nonconvex constraints up to a number of violations bounded by a linear function of the state dimension~$n_x$, where the exact form of the bound may vary across different problems. The second part addresses trajectory generation under general nonconvex constraints. We first introduce a unified modeling and algorithmic framework that integrates prox-linear methods with exact penalty formulations. Moreover, due to discretization, classical trajectory generation algorithms typically guarantee constraint satisfaction only at grid points, and violations may inevitably occur between grid points. To address this issue, we propose a novel approach that incorporates an integral reformulation of the constraints into the optimization procedure, thereby ensuring constraint satisfaction over the entire time horizon. Lastly, to reduce the manual effort commonly required for parameter tuning, we design a proportional-integral (PI)-inspired autotuning scheme within this framework, which introduces a vectorized exact penalty comprising both linear and quadratic terms. After each prox-linear subproblem is solved, the penalty weights are adaptively updated: the linear term accumulates constraint violations across iterations, analogous to the integral part of PI, while the quadratic term responds directly to the current violation, corresponding to the proportional part. Theoretical guarantees and convergence analysis are provided for all methods introduced in this part. Finally, we propose Newton-PIPG, an efficient method for solving quadratic programming (QP) problems arising in optimal control, subject to additional set constraints. Such problems can serve as subproblems from general nonconvex trajectory generation algorithms. Newton-PIPG integrates the Proportional-Integral Projected Gradient (PIPG) method with the Newton method, thereby achieving both global convergence and local quadratic convergence. The PIPG method, an operator-splitting algorithm, seeks a fixed point of the PIPG operator. Under mild assumptions, we demonstrate that this operator is locally smooth, which enables the application of the Newton method to solve the corresponding nonlinear fixed-point equation. Furthermore, we prove that the linear system associated with the Newton method is locally nonsingular under strict complementarity conditions. To enhance computational efficiency, we developed a specialized matrix factorization technique that exploits the typical sparsity structure of optimal control problems and makes use of block Cholesky decomposition. Numerical experiments demonstrate that Newton-PIPG achieves high accuracy and reduces computation time, particularly in settings where feasibility is easily guaranteed.