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

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.

💡 Research Summary

This paper introduces a novel inner‑approximation algorithm for dynamic optimization problems with continuous‑time path constraints, guaranteeing strict feasibility throughout the entire horizon. The method combines a q‑th order Taylor expansion of each constraint on a sub‑interval with the convex‑hull property of Bernstein polynomials. After normalizing the time variable to the unit interval, the polynomial part of the Taylor series is expressed in Bernstein form, allowing the maximum Bernstein coefficient to serve as a tight upper bound for the polynomial component. The remainder term, which captures higher‑order effects, is bounded using global interval enclosures of the q‑th derivative, yielding a conservative but analytically tractable bound R_q(T).

To avoid the nondifferentiability of the max operator, the authors replace it with a Log‑Sum‑Exp (LSE) smoothing function:

H_TB(u,T) = (1/ρ)·ln(∑_{j=0}^r exp(ρ·b_j(u,T))) + R_q(T),

where ρ>0 is a smoothing parameter and b_j(u,T) are the Bernstein coefficients. Theorem 1 proves that H_TB is a strict upper bound for the original constraint h(u,t) on any sub‑interval T, with an over‑estimation error bounded by δ = ln(r+1)/ρ plus terms of order Δ(T)^2 and Δ(T)^q.

The algorithm proceeds iteratively: (1) partition the time horizon, (2) solve a nonlinear programming (NLP) problem that includes the smooth upper‑bound constraints H_TB(u,T) ≤ 0, (3) evaluate the actual constraint violation on each interval, (4) refine intervals where violations are detected or increase ρ to tighten the bound, and (5) repeat until all intervals satisfy the smooth bound. The authors prove finite‑step convergence to a Karush‑Kuhn‑Tucker (KKT) point of the original problem while maintaining strict feasibility at every iteration.

Numerical experiments on three benchmark problems—chemical process optimization, robotic trajectory planning, and model predictive control—demonstrate substantial improvements over traditional interval‑analysis‑based inner‑approximation methods. The proposed algorithm reduces the number of constraints by roughly 30–50 % and cuts total solution time by 45–60 %, while never violating the original path constraints. The results confirm that the Bernstein‑based envelope effectively suppresses dependency‑induced over‑estimation, and the LSE smoothing preserves gradient information for efficient NLP solving.

In summary, the Taylor‑Bernstein inner‑approximation framework offers a mathematically rigorous, computationally efficient way to handle infinite‑dimensional path constraints in dynamic optimization. Future work will explore adaptive strategies for the smoothing parameter, extensions to multi‑objective settings, and data‑driven techniques for estimating derivative bounds without explicit analytical expressions.