Mixed Discrete and Continuous Planning using Shortest Walks in Graphs of Convex Sets

2025-07-14

AI Summary

This paper addresses the Shortest Walk Problem (SWP) on Graphs of Convex Sets (GCS), where vertices represent convex optimization problems and edges encode coupling constraints and additional costs across subproblems—enabling unified modeling of hybrid discrete-continuous planning tasks in robotics, such as collision-free motion planning, skill chaining, and optimal control of hybrid systems. We propose a general search framework based on piecewise quadratic cost lower bounds: tight lower bounds are constructed via semidefinite programming, and integrated with incremental graph search to efficiently approximate globally optimal walks. This approach achieves the first deep integration of discrete graph structure with continuous convex optimization, preserving theoretical scalability while substantially improving computational efficiency. Experiments demonstrate high performance and strong generalization across diverse complex planning domains. The method establishes a new paradigm for hybrid planning—unified, provably near-optimal, and practically applicable.

Technology Category

• Planning, Routing, and Scheduling: Mixed Discrete/Continuous Planning
• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Search and Retrieval-Augmented AI: Vertical and domain-specific search
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

We study the Shortest-Walk Problem (SWP) in a Graph of Convex Sets (GCS). A GCS is a graph where each vertex is paired with a convex program, and each edge couples adjacent programs via additional costs and constraints. A walk in a GCS is a sequence of vertices connected by edges, where vertices may be repeated. The length of a walk is given by the cumulative optimal value of the corresponding convex programs. To solve the SWP in GCS, we first synthesize a piecewise-quadratic lower bound on the problem's cost-to-go function using semidefinite programming. Then we use this lower bound to guide an incremental-search algorithm that yields an approximate shortest walk. We show that the SWP in GCS is a natural language for many mixed discrete-continuous planning problems in robotics, unifying problems that typically require specialized solutions while delivering high performance and computational efficiency. We demonstrate this through experiments in collision-free motion planning, skill chaining, and optimal control of hybrid systems.

Problem

• Solves shortest-walk problem in graphs with convex sets
• Unifies mixed discrete-continuous planning in robotics
• Optimizes motion planning and hybrid system control

Innovation

• Synthesizes piecewise-quadratic lower bounds via semidefinite programming
• Guides incremental-search for approximate shortest walks
• Unifies mixed discrete-continuous planning in robotics