Algorithms for Nonlinear Mixed-Integer Location Estimation

2025-05-19

AI Summary

This paper addresses the nonlinear mixed-integer least-squares positioning problem—featuring continuous position variables and discrete integer ambiguities—in GNSS-denied environments (e.g., terrestrial and indoor scenarios). Conventional linearization-based methods fail under short-range, highly nonlinear conditions; to overcome this limitation, we propose two novel algorithms: (1) a constraint elimination method that analytically removes nonlinear terms prior to optimization, reducing the original problem to a pure integer optimization; and (2) a geometry-driven polynomial-time enumeration method that constructs a finite, complete integer solution space leveraging ranging geometry. Integrating nonlinear optimization, integer least squares, and geometric modeling, our approach achieves significantly higher integer ambiguity resolution success rates and positioning accuracy than linearized baselines in simulations—enabling high-precision, tightly coupled short-range positioning without GNSS.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

For three decades, carrier-phase observations have been used to obtain the most accurate location estimates using global navigation satellite systems (GNSS). These estimates are computed by minimizing a nonlinear mixed-integer least-squares problem. Existing algorithms linearize the problem, orthogonally project it to eliminate real variables, and then solve the integer least-square problem. There is now considerable interest in developing similar localization techniques for terrestrial and indoor settings. We show that algorithms that linearize first fail in these settings and we propose several algorithms for computing the estimates. Some of our algorithms are elimination algorithms that start by eliminating the non-linear terms in the constraints; others construct a geometric arrangement that allows us to efficiently enumerate integer solutions (in polynomial time). We focus on simplified localization problems in which the measurements are range (distance) measurements and carrier phase range measurements, with no nuisance parameters. The simplified problem allows us to focus on the core question of untangling the nonlinearity and the integer nature of some parameters. We show using simulations that the new algorithms are effective at close ranges at which the linearize-first approach fails.

Problem

• Solving nonlinear mixed-integer GNSS location estimation problems
• Developing terrestrial and indoor localization techniques effectively
• Eliminating nonlinear terms and efficiently enumerating integer solutions

Innovation

• Eliminate nonlinear terms in constraints first
• Construct geometric arrangement for integer solutions
• Focus on range and carrier phase measurements