Frank-Wolfe Recursions for the Emergency Response Problem on Measure Spaces

2025-07-13

AI Summary

This study addresses the spatial resource allocation of volunteer responders in out-of-hospital cardiac arrest (OHCA) emergency response, formulating an infinite-dimensional optimization model on a measure space to minimize patient mortality probability. We propose a fully corrected Frank–Wolfe algorithm that avoids discretization or parametric approximation and permits inexact subproblem solutions. Leveraging influence functions as first-order variational tools and modeling volunteer relocation paths via the L₁ norm, the algorithm enables adaptive support set reduction. Theoretically, we characterize nontrivial structural properties of optimal spatial resource configurations in the continuous setting, extending the classical P-median framework. Evaluated on real-world OHCA data from Oakland, the method scales to large urban environments, achieving significant improvements in response time efficiency and patient survival rates.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

We consider an optimization problem over measures for emergency response to out-of-hospital cardiac arrest (OHCA), where the goal is to allocate volunteer resources across a spatial region to minimize the probability of death. The problem is infinite-dimensional and poses challenges for analysis and computation. We first establish structural properties, including convexity of the objective functional, compactness of the feasible set, and existence of optimal solutions. We also derive the influence function, which serves as the first-order variational object in our optimization framework. We then adapt and analyze a fully-corrective Frank-Wolfe (fc-FW) algorithm that operates directly on the infinite-dimensional problem without discretization or parametric approximation. We show a form of convergence even when subproblems are not solved to global optimality. Our full implementation of fc-FW demonstrates complex solution structure even in simple discrete cases, reveals nontrivial volunteer allocations in continuous cases, and scales to realistic urban scenarios using OHCA data from the city of Auckland, New Zealand. Finally, we show that when volunteer travel is modeled through the $L_1$ norm, the influence function is piecewise strictly concave, enabling fast computation via support reduction. The proposed framework and analysis extend naturally to a broad class of $P$-means problems.

Problem

• Optimize volunteer allocation to minimize OHCA death probability
• Solve infinite-dimensional measure-space problem without discretization
• Analyze Frank-Wolfe algorithm convergence for complex spatial solutions

Innovation

• Fully-corrective Frank-Wolfe algorithm without discretization
• Influence function for first-order variational optimization
• Fast computation via support reduction for L1 norm