A Taylor-Bernstein Inner Approximation Algorithm for Path-Constrained Dynamic Optimization

TL;DR

This paper introduces a new inner approximation algorithm for path-constrained dynamic optimization that uses Bernstein polynomial properties and smoothing techniques to improve accuracy and computational efficiency while guaranteeing strict feasibility.

Contribution

The paper presents a novel algorithm leveraging Bernstein polynomial convex hulls and Log-Sum-Exp smoothing for tighter bounds in dynamic optimization with path constraints.

Findings

Algorithm converges in finite steps to a KKT solution.

Reduces number of constraints in the approximation problem.

Improves computational performance while ensuring strict feasibility.

Abstract

A novel inner approximation algorithm is proposed for dynamic optimization problems to ensure strict satisfaction of path constraints. Distinct from traditional methods relying on interval analysis, the proposed algorithm leverages the convex hull property of Bernstein polynomials to tightly bound the polynomial components of the Taylor expansion, while incorporating the Log-Sum-Exp technique to smooth the non-differentiability arising from coefficient maximization. This approach yields a tighter upper bound function compared to interval methods, with a smaller approximation error. Theoretical analysis shows that the algorithm converges in a finite number of steps to a KKT solution of the original problem that satisfies the specified tolerances. Numerical simulations confirm that the proposed algorithm effectively reduces the number of constraints in the approximation problem, improving computational performance while ensuring strict feasibility.