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Tight Bounds on Polynomials and Its Application to Dynamic Optimization Problems

Eduardo M. G. Vila, Eric C. Kerrigan, Paul Bruce

TL;DR

The paper addresses enforcing tight, global polynomial bounds within dynamic optimization problems solved by pseudo-spectral methods. It combines Bernstein polynomial constraints with flexible sub-interval discretizations to tightly bound interpolants while maintaining dynamics. Key contributions include proving that a finite number of sub-intervals suffices for tight bounds on piecewise monotone polynomials and demonstrating up to a tenfold improvement in relative cost on Bryson–Denham and cart-pole examples. The framework improves constraint satisfaction without sacrificing spectral convergence, offering a scalable approach for DOPs with general inequality constraints.

Abstract

This paper presents a pseudo-spectral method for Dynamic Optimization Problems (DOPs) that allows for tight polynomial bounds to be achieved via flexible sub-intervals. The proposed method not only rigorously enforces inequality constraints, but also allows for a lower cost in comparison with non-flexible discretizations. Two examples are provided to demonstrate the feasibility of the proposed method to solve optimal control problems. Solutions to the example problems exhibited up to a tenfold reduction in relative cost.

Tight Bounds on Polynomials and Its Application to Dynamic Optimization Problems

TL;DR

The paper addresses enforcing tight, global polynomial bounds within dynamic optimization problems solved by pseudo-spectral methods. It combines Bernstein polynomial constraints with flexible sub-interval discretizations to tightly bound interpolants while maintaining dynamics. Key contributions include proving that a finite number of sub-intervals suffices for tight bounds on piecewise monotone polynomials and demonstrating up to a tenfold improvement in relative cost on Bryson–Denham and cart-pole examples. The framework improves constraint satisfaction without sacrificing spectral convergence, offering a scalable approach for DOPs with general inequality constraints.

Abstract

This paper presents a pseudo-spectral method for Dynamic Optimization Problems (DOPs) that allows for tight polynomial bounds to be achieved via flexible sub-intervals. The proposed method not only rigorously enforces inequality constraints, but also allows for a lower cost in comparison with non-flexible discretizations. Two examples are provided to demonstrate the feasibility of the proposed method to solve optimal control problems. Solutions to the example problems exhibited up to a tenfold reduction in relative cost.
Paper Structure (31 sections, 3 theorems, 49 equations, 11 figures)

This paper contains 31 sections, 3 theorems, 49 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Constraining Polynomials
  3. Achieving Tight Bounds
  4. Dynamic Optimization
  5. Contributions and Outline
  6. Tightly Bounded Polynomials
  7. The Bernstein Basis
  8. Monotonicity
  9. Finite Number of Sub-Intervals
  10. Dynamic Optimization Problems
  11. Flexible Discretizations
  12. Flexible Sub-Intervals
  13. Dynamic Variables
  14. Cost Function
  15. Equality Constraints
  16. ...and 16 more sections

Key Result

Lemma 1

Let $p_{\min}$ and $p_{\max}$ be the minimum and maximum values of $p(t)$$\forall t \in [0, 1]$. In $[0, 1]$, $p$ is bounded by Moreover, the bounds are tight in the following cases:

Figures (11)

  • Figure 1: Examples of 3rd degree polynomials and their Bernstein coefficients $\beta_j$.
  • Figure 2: Bernstein-constrained piecewise polynomial approximations with equispaced and flexed sub-intervals.
  • Figure 3: Approximation errors for equispaced and flexed sub-intervals, with and without Bernstein constraints.
  • Figure 4: Bernstein basis polynomials of degree 4, along with their sum.
  • Figure 5: Examples of monotonic polynomials that are not tightly bounded by their Bernstein coefficients. The polynomials are given in Appendix A.
  • ...and 6 more figures

Theorems & Definitions (5)

  • Lemma 1
  • Lemma 2
  • proof
  • Theorem 1
  • proof