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Hidden convexity property of a speed planning problem

Stefano Ardizzoni, Luca Consolini, Mattia Laurini, Marco Locatelli

Abstract

In this paper we address the speed planning problem for a vehicle along a predefined path. A weighted average of two (conflicting) terms, energy consumption and travel time, is minimized. After deriving a non-convex mathematical model of the problem, we introduce a convex relaxation of the model and show that, after the application of a suitable feasibility-based bound tightening procedure, the convex relaxation shares the same optimal value and solution of the non-convex problem. We also establish that the feasible region of the non-convex problem is a lattice and, through that, a necessary and sufficient condition for the non-emptiness of the feasible region.

Hidden convexity property of a speed planning problem

Abstract

In this paper we address the speed planning problem for a vehicle along a predefined path. A weighted average of two (conflicting) terms, energy consumption and travel time, is minimized. After deriving a non-convex mathematical model of the problem, we introduce a convex relaxation of the model and show that, after the application of a suitable feasibility-based bound tightening procedure, the convex relaxation shares the same optimal value and solution of the non-convex problem. We also establish that the feasible region of the non-convex problem is a lattice and, through that, a necessary and sufficient condition for the non-emptiness of the feasible region.

Paper Structure

This paper contains 12 sections, 9 theorems, 52 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Minimum-time speed planning
  4. Fuel consumption minimization
  5. Outline of the paper
  6. Convex relaxation of the problem
  7. Exactness result for the case without final boundary condition
  8. Feasibility-based bound tightening
  9. Exact convex relaxation with final boundary condition
  10. Further bound-tightening techniques
  11. Lattice structure and a necessary and sufficient condition for feasibility
  12. Conclusions

Key Result

Proposition 3.3

Let us assume that Assumptions assum:1 and assum:2 hold. Then, the optimal solution of the relaxed problem (eq:relaxfix) is feasible and, thus, optimal for problem eq:probfix if we remove constraint $w_n=w_{\text{fin}}$ in both problems.

Theorems & Definitions (9)

  • Proposition 3.3
  • Lemma 4.2
  • Lemma 4.3
  • Proposition 4.4
  • Proposition 4.5
  • Proposition 5.1
  • Proposition 7.1
  • Proposition 7.2
  • Proposition 7.3