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A Sequential Operator-Splitting Framework for Exploration of Nonconvex Trajectory Optimization Solution Spaces

Justin Ganiban, Natalia Pavlasek, Behcet Acikmese

TL;DR

The paper addresses the tendency of standard SCP to converge to local minima in nonconvex trajectory optimization by introducing OS-SCP, a multi-agent extension that uses diverse initial trajectories coupled via consensus ADMM. Each agent solves a convex subproblem with a modified penalized objective, followed by a consensus projection onto a convexified constraint set and a dual update to maintain coherence among agents. Empirical results on a unicycle model with and without Gaussian terrain bias demonstrate that OS-SCP can achieve equal or lower costs than standard SCP initialized with the same guesses, often with similar computational effort, and without manual selection of a near-optimal initial trajectory. The approach provides a practical means to explore nonconvex solution spaces, reduce the need for careful initialization, and potentially improve reliability in safety-critical trajectory generation tasks.

Abstract

Trajectory optimization methods provide an efficient and reliable means of computing feasible trajectories in nonconvex solution spaces. However, a well-known limitation of these algorithms is that they are inherently local in nature, and typically converge to a solution in the neighborhood of their initial guess. This paper presents a sequential operator-splitting framework, based on the alternating direction method of multipliers (ADMM), aimed at promoting exploration within the sequential convex programming (SCP) framework. In particular, diverse initial solutions are modeled as agents within the consensus ADMM framework. Driving these agents toward consensus facilitates exploration of the nonconvex optimization landscape. Numerical simulations demonstrate that the proposed method consistently yields equivalent or lower-cost solutions compared to the standard SCP approach, with the same number of or fewer agents.

A Sequential Operator-Splitting Framework for Exploration of Nonconvex Trajectory Optimization Solution Spaces

TL;DR

The paper addresses the tendency of standard SCP to converge to local minima in nonconvex trajectory optimization by introducing OS-SCP, a multi-agent extension that uses diverse initial trajectories coupled via consensus ADMM. Each agent solves a convex subproblem with a modified penalized objective, followed by a consensus projection onto a convexified constraint set and a dual update to maintain coherence among agents. Empirical results on a unicycle model with and without Gaussian terrain bias demonstrate that OS-SCP can achieve equal or lower costs than standard SCP initialized with the same guesses, often with similar computational effort, and without manual selection of a near-optimal initial trajectory. The approach provides a practical means to explore nonconvex solution spaces, reduce the need for careful initialization, and potentially improve reliability in safety-critical trajectory generation tasks.

Abstract

Trajectory optimization methods provide an efficient and reliable means of computing feasible trajectories in nonconvex solution spaces. However, a well-known limitation of these algorithms is that they are inherently local in nature, and typically converge to a solution in the neighborhood of their initial guess. This paper presents a sequential operator-splitting framework, based on the alternating direction method of multipliers (ADMM), aimed at promoting exploration within the sequential convex programming (SCP) framework. In particular, diverse initial solutions are modeled as agents within the consensus ADMM framework. Driving these agents toward consensus facilitates exploration of the nonconvex optimization landscape. Numerical simulations demonstrate that the proposed method consistently yields equivalent or lower-cost solutions compared to the standard SCP approach, with the same number of or fewer agents.

Paper Structure

This paper contains 17 sections, 1 theorem, 23 equations, 9 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Consensus ADMM
  4. Sequential Convex Programming
  5. Exploratory SCP Algorithm
  6. Operator-Splitting SCP (OS-SCP)
  7. Primal update
  8. Consensus update
  9. Dual update
  10. Numerical Results
  11. Unicycle Trajectory Optimization
  12. SCP solution
  13. OS-SCP solution
  14. Unicycle Trajectory Optimization with Gaussian Terrain Fields
  15. SCP Solution
  16. ...and 2 more sections

Key Result

Proposition II.1

If each $p_i$ and $q$ are convex, closed, and proper, then the consensus residuals vanish, meaning that as $j \rightarrow \infty$, the primal residual Moreover, the dual residuals vanish as $j \rightarrow \infty$ such that

Figures (9)

  • Figure 1: Overview of standard SCP and OS-SCP. Initial state is represented by a black circle and goal state by black star. Two obstacles are shown in white.
  • Figure 2: Problem setup for simple obstacle avoidance example.
  • Figure 3: Initial and converged trajectories computed using standard SCP algorithm.
  • Figure 4: Initial and converged trajectories computed using OS-SCP algorithm.
  • Figure 5: Primal and dual residuals over iterations for OS-SCP.
  • ...and 4 more figures

Theorems & Definitions (2)

  • Proposition II.1
  • proof