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Dynamic Programming Algorithms for Finding Cost-Optimal Trajectory on the Terrain

Majid E. Abbasov, Anna A. Gorbunova

TL;DR

The paper addresses cost-optimal trajectory design on a terrain by formulating a 3D calculus-of-variations problem with a composite cost functional $J(y)$ that accounts for material delivery and labor costs along with terrain elevation. It develops a dynamic-programming scheme based on Bellman's principle, proves the existence of a minimizer, and establishes convergence of the discretized algorithm under a refined grid strategy $\Delta_k=\gamma\tau_k^{1+\varepsilon}$, with a quantified complexity bound. A local-search modification is proposed to reduce computational burden at the expense of global optimality, and a 2D simplification is offered when elevation changes are negligible. Numerical experiments benchmark the DP approaches against Ritz-method solutions, illustrating trade-offs between accuracy and efficiency and demonstrating applicability to constrained problems by excluding forbidden regions.

Abstract

This paper considers the problem of finding the optimal trajectory of a road on a given terrain, taking into account the presence of elevation changes. Unlike the heuristic methods often used to solve such problems we present an approach that reduces the problem to a calculus of variations problem by introducing an integral cost functional. We prove the existence of an optimum of the functional. The optimal trajectory that minimizes the specified functional is sought using the dynamic programming method. We also thoroughly study the discretization issue and demonstrate the implementation of Bellman's optimality principle. In addition, we prove the convergence theorem of the proposed algorithm and provide a computational complexity analysis as well as numerical examples.

Dynamic Programming Algorithms for Finding Cost-Optimal Trajectory on the Terrain

TL;DR

The paper addresses cost-optimal trajectory design on a terrain by formulating a 3D calculus-of-variations problem with a composite cost functional that accounts for material delivery and labor costs along with terrain elevation. It develops a dynamic-programming scheme based on Bellman's principle, proves the existence of a minimizer, and establishes convergence of the discretized algorithm under a refined grid strategy , with a quantified complexity bound. A local-search modification is proposed to reduce computational burden at the expense of global optimality, and a 2D simplification is offered when elevation changes are negligible. Numerical experiments benchmark the DP approaches against Ritz-method solutions, illustrating trade-offs between accuracy and efficiency and demonstrating applicability to constrained problems by excluding forbidden regions.

Abstract

This paper considers the problem of finding the optimal trajectory of a road on a given terrain, taking into account the presence of elevation changes. Unlike the heuristic methods often used to solve such problems we present an approach that reduces the problem to a calculus of variations problem by introducing an integral cost functional. We prove the existence of an optimum of the functional. The optimal trajectory that minimizes the specified functional is sought using the dynamic programming method. We also thoroughly study the discretization issue and demonstrate the implementation of Bellman's optimality principle. In addition, we prove the convergence theorem of the proposed algorithm and provide a computational complexity analysis as well as numerical examples.

Paper Structure

This paper contains 9 sections, 5 theorems, 65 equations, 4 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Problem Statement and Formalization
  3. Dynamic Programming Scheme
  4. Existence of the Solution and Convergence of the Algorithm
  5. Local Search Modification
  6. Modification for the Local Search
  7. 2D Simplification of the Model
  8. Numerical Experiments
  9. Conclusion

Key Result

Theorem 1

A family of continuously differentiable functions $Y$ defined on $[0,l]$ is compact in $\mathbb{C}^1[0,l]$ iff $Y$ is uniformly bounded and equicontinuous in $\mathbb{C}^1[0,l]$.

Figures (4)

  • Figure 1: An illustration of the dynamic programming algorithm for the considered problem
  • Figure 2: An illustration of the algorithm for finding a local solution. Circles indicate feasible points while crosses indicate infeasible points
  • Figure 3: A graph of the optimal solution in Example \ref{['AM_GA_exmpl1']}
  • Figure 4: A graph of the optimal solution in Example \ref{['AM_GA_exmpl2']}

Theorems & Definitions (11)

  • Theorem : Arzela-Ascoli
  • Theorem : Existence of a solution
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • Remark 1
  • Example 1
  • ...and 1 more