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

Yuan Chang, Lizhong Jiang, Tai-Fang Li, Jun Fu

TL;DR

This work tackles path-constrained dynamic optimization by introducing a Taylor-Bernstein inner approximation that leverages Bernstein polynomial convex hulls to bound the Taylor expansion of path constraints and uses Log-Sum-Exp smoothing to obtain a differentiable, strictly conservative upper bound. The authors prove finite convergence to a KKT point of the original problem and demonstrate, through three benchmarks, that the method significantly reduces the number of constraints and lowers computation time (approximately 30–40% faster) while guaranteeing strict feasibility. Key innovations include a rigorous upper bound bound $H_{TB}$ on path constraints, a gradient-friendly formulation via a precomputed transform $M(T)$, and an adaptive subdivision strategy that ensures finite termination under mild assumptions. The approach offers practical gains for large-scale dynamic optimization with infinite-dimensional constraints and sets the stage for applying Taylor-Bernstein techniques to non-smooth or closed-loop settings.

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.

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

TL;DR

This work tackles path-constrained dynamic optimization by introducing a Taylor-Bernstein inner approximation that leverages Bernstein polynomial convex hulls to bound the Taylor expansion of path constraints and uses Log-Sum-Exp smoothing to obtain a differentiable, strictly conservative upper bound. The authors prove finite convergence to a KKT point of the original problem and demonstrate, through three benchmarks, that the method significantly reduces the number of constraints and lowers computation time (approximately 30–40% faster) while guaranteeing strict feasibility. Key innovations include a rigorous upper bound bound on path constraints, a gradient-friendly formulation via a precomputed transform , and an adaptive subdivision strategy that ensures finite termination under mild assumptions. The approach offers practical gains for large-scale dynamic optimization with infinite-dimensional constraints and sets the stage for applying Taylor-Bernstein techniques to non-smooth or closed-loop settings.

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.
Paper Structure (13 sections, 2 theorems, 45 equations, 6 figures, 1 table, 1 algorithm)

This paper contains 13 sections, 2 theorems, 45 equations, 6 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Statement and Preliminaries
  3. Problem Statement
  4. Preliminaries
  5. Construction of Path Constraint Upper Bound Functions
  6. Upper Bound Function Construction
  7. Gradient Analysis
  8. Algorithm Design and Convergence Analysis
  9. Algorithm Description
  10. Convergence Analysis
  11. Simulation Examples
  12. Conclusion
  13. Numerical Examples

Key Result

Theorem 1

For any smoothing parameter $\rho > 0$, polynomial degree $r \geqslant 1$, and time subinterval $T \subseteq I$, the smooth function $H_{TB}(u, T)$ defined in (17) serves as a strict upper bound for the path constraint function $h(u, t)$. Specifically, the overestimation error gap, defined as $E(u,

Figures (6)

  • Figure 1: Comparison of polynomial enclosures between standard interval arithmetic and the Bernstein form. This figure illustrates four different polynomial functions, where shaded areas represent the respective enclosures. In the first three cases, the bounds of standard interval arithmetic are omitted from the view as they significantly exceed the vertical axis limits due to severe overestimation. In contrast, the Bernstein form yields significantly tighter bounds by leveraging the convex hull property, demonstrating its capability to suppress the dependency effect and reduce conservatism.
  • Figure 2: Algorithm flowchart
  • Figure 3: Simulation results of Example 1.
  • Figure 4: Simulation results of Example 2.
  • Figure 5: Simulation results of Example 3.
  • ...and 1 more figures

Theorems & Definitions (8)

  • Remark 1
  • Theorem 1
  • proof
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 2
  • proof