凸集合图中的最短路径

IF 2.6 1区数学 Q1 MATHEMATICS, APPLIED

SIAM Journal on Optimization Pub Date : 2024-02-01 DOI:10.1137/22m1523790

Tobia Marcucci, Jack Umenberger, Pablo Parrilo, Russ Tedrake

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引用次数: 0

摘要

SIAM 优化期刊》，第 34 卷第 1 期，第 507-532 页，2024 年 3 月。摘要给定一个图，最短路径问题要求找到一个连接源顶点和目标顶点的累积长度最小的边序列。我们考虑了这一经典问题的一个变体，其中图中每个顶点的位置是一个连续的决策变量，受限于一个凸集，而边的长度是其端点位置的凸函数。从自动驾驶汽车的运动规划到混合动力系统的优化控制，这种形式的问题自然出现在许多领域。这种广泛适用性的代价是这一问题的复杂性，很容易被认为是 NP 难。我们的主要贡献是基于透视算子的强大而轻便的混合整数凸表述，它使得在大型图和高维空间中高效地找到全局最优路径成为可能。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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Shortest Paths in Graphs of Convex Sets

SIAM Journal on Optimization, Volume 34, Issue 1, Page 507-532, March 2024.
Abstract. Given a graph, the shortest-path problem requires finding a sequence of edges with minimum cumulative length that connects a source vertex to a target vertex. We consider a variant of this classical problem in which the position of each vertex in the graph is a continuous decision variable constrained in a convex set, and the length of an edge is a convex function of the position of its endpoints. Problems of this form arise naturally in many areas, from motion planning of autonomous vehicles to optimal control of hybrid systems. The price for such a wide applicability is the complexity of this problem, which is easily seen to be NP-hard. Our main contribution is a strong and lightweight mixed-integer convex formulation based on perspective operators, that makes it possible to efficiently find globally optimal paths in large graphs and in high-dimensional spaces.

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来源期刊

SIAM Journal on Optimization 数学-应用数学

CiteScore

5.30

自引率

9.70%

发文量

101

审稿时长

6-12 weeks

期刊介绍： The SIAM Journal on Optimization contains research articles on the theory and practice of optimization. The areas addressed include linear and quadratic programming, convex programming, nonlinear programming, complementarity problems, stochastic optimization, combinatorial optimization, integer programming, and convex, nonsmooth and variational analysis. Contributions may emphasize optimization theory, algorithms, software, computational practice, applications, or the links between these subjects.