Automatic Discovery of Redundant Constraints for Globally Optimal State Estimation

Core Concepts

This paper proposes an automatic method to find a set of sufficient redundant constraints to obtain tight semidefinite relaxations for globally optimal solutions to state estimation problems in robotics.

Abstract

The paper presents two methods, AUTOTIGHT and AUTOTEMPLATE, to automatically discover redundant constraints required for tightening semidefinite relaxations of common optimization problems in robotics, such as range-based localization and stereo-based pose estimation.
AUTOTIGHT determines if a given problem formulation can be tightened by adding enough redundant constraints, without requiring any manual steps for guessing the constraints. It does this by numerically retrieving the nullspace basis of a data matrix constructed from randomly generated feasible samples.
AUTOTEMPLATE builds on AUTOTIGHT to automatically determine a set of 'constraint templates' that can be generalized to problems of any size. This is achieved by identifying different variable types in the problem and learning templates that can be applied to any combination of these variables.
The proposed methods circumvent the tedious manual process typically required to find the right redundant constraints for tightening semidefinite relaxations. This lowers the barrier for adopting semidefinite programming techniques to find globally optimal solutions to optimization problems in robotics.
The effectiveness of the approach is showcased in simulation and on real datasets for range-based localization and stereo-based pose estimation. The authors also reproduce semidefinite relaxations from recent literature and show that their automatic method always finds a smaller set of constraints sufficient for tightness than previously considered.

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Key Insights Distilled From

Toward Globally Optimal State Estimation Using Automatically Tightened Semidefinite Relaxations

by Fred... at arxiv.org 04-16-2024

https://arxiv.org/pdf/2308.05783.pdf

Deeper Inquiries

How can the proposed methods be extended to handle problems with non-quadratic objective functions or more complex constraint structures

The proposed methods, AUTOTIGHT and AUTOTEMPLATE, can be extended to handle problems with non-quadratic objective functions or more complex constraint structures by incorporating higher-order terms in the lifting process. In the context of semidefinite relaxations, the lifting process involves adding polynomial terms of increasing order to the original problem to create a higher-dimensional SDP. By including higher-order terms in the lifting process, the relaxation can capture non-quadratic objective functions and more complex constraint structures. This extension allows for a more flexible representation of the optimization problem and enables the methods to handle a wider range of problem formulations.

What are the limitations of the sampling-based approach used in AUTOTIGHT and AUTOTEMPLATE, and how could they be addressed

The sampling-based approach used in AUTOTIGHT and AUTOTEMPLATE has certain limitations that need to be addressed. One limitation is the reliance on randomly generated training data to determine the nullspace basis or constraint templates. This approach may not always capture the full complexity of the problem space, especially in high-dimensional or non-linear optimization problems. To address this limitation, one possible solution is to incorporate domain knowledge or problem-specific information into the sampling process. This can help ensure that the training data adequately represents the problem space and leads to more accurate and reliable results.
Another limitation is the scalability of the sampling-based approach, especially when dealing with large problem sizes. Generating a sufficient number of feasible samples to ensure the rank-deficiency of the data matrix can be computationally expensive and time-consuming. To improve scalability, techniques such as adaptive sampling strategies or dimensionality reduction methods could be employed to reduce the computational burden while still maintaining the effectiveness of the approach. Additionally, exploring alternative sampling techniques or optimization algorithms that are better suited for high-dimensional problems could help address the scalability limitations of the sampling-based approach.

Could the learned constraint templates be used to guide the design of new problem formulations that are more amenable to tight semidefinite relaxations

The learned constraint templates can be used to guide the design of new problem formulations that are more amenable to tight semidefinite relaxations by providing insights into the structure and relationships within the optimization problem. By analyzing the learned templates, researchers can identify common patterns or constraints that contribute to the tightness of the relaxation. This knowledge can then be used to inform the design of new problem formulations that incorporate similar constraints or structures, leading to tighter relaxations and potentially better optimization performance.
Furthermore, the constraint templates can serve as a starting point for developing customized optimization algorithms or problem-specific solvers that leverage the identified patterns to improve efficiency and effectiveness. By incorporating the learned constraints into the formulation process, researchers can iteratively refine and optimize the problem representation to enhance the overall performance of the optimization process. Overall, the learned constraint templates offer valuable insights that can drive the development of more tailored and effective problem formulations for semidefinite relaxation-based optimization problems.