Vectorized Sparse Second-Order Forward Automatic Differentiation for Optimal Control Direct Methods

TL;DR

This paper introduces a vectorized sparse second-order forward automatic differentiation framework tailored for direct collocation methods in optimal control, significantly improving the efficiency of derivative computations by exploiting problem sparsity.

Contribution

The paper presents a novel framework that combines vectorization and sparsity exploitation for second-order automatic differentiation in optimal control, enabling faster and more scalable computations.

Findings

Efficient derivative computation across multiple mesh points.

Enhanced parallelization and memory access patterns.

Open-source implementation supporting complex multi-phase problems.

Abstract

Direct collocation methods are widely used numerical techniques for solving optimal control problems. The discretization of continuous-time optimal control problems transforms them into large-scale nonlinear programming problems, which require efficient computation of first- and second-order derivatives. To achieve computational efficiency, these derivatives must be computed in sparse and vectorized form, exploiting the problem's inherent sparsity structure. This paper presents a vectorized sparse second-order forward automatic differentiation framework designed for direct collocation methods in optimal control. The method exploits the problem's sparse structure to efficiently compute derivatives across multiple mesh points. By incorporating both scalar and vector nodes within the expression graph, the approach enables effective parallelization and optimized memory access patterns while maintaining flexibility for complex problems. The methodology is demonstrated through application to a prototype optimal control problem. A complete implementation for multi-phase optimal control problems is available as an open-source package, supporting both theoretical research and practical applications.