GaussNewtonOptimizer

Overview¶

The GaussNewtonOptimizer class in GTSAM is designed to optimize nonlinear factor graphs using the Gauss-Newton algorithm. This class is particularly suited for problems where the cost function can be approximated well by a quadratic function near the minimum. The Gauss-Newton method is an iterative optimization technique that updates the solution by linearizing the nonlinear system at each iteration.

The Gauss-Newton algorithm is based on the idea of linearizing the nonlinear residuals $r(x)$ around the current estimate $x_k$ . The update step is derived from solving the normal equations:

J(x_k)^T J(x_k) \Delta x = -J(x_k)^T r(x_k)

(1)

where $J(x_k)$ is the Jacobian of the residuals with respect to the variables. The solution $\Delta x$ is used to update the estimate:

x_{k+1} = x_k + \Delta x

(2)

This process is repeated iteratively until convergence.

Key features:

Iterative Optimization: The optimizer refines the solution iteratively by linearizing the nonlinear system around the current estimate.
Convergence Control: It provides mechanisms to control the convergence through parameters such as maximum iterations and relative error tolerance.
Integration with GTSAM: Seamlessly integrates with GTSAM’s factor graph framework, allowing it to be used with various types of factors and variables.

Key Methods¶

Please see the base class NonlinearOptimizer.

Parameters¶

The Gauss-Newton optimizer uses the standard optimization parameters inherited from NonlinearOptimizerParams, which include:

Maximum iterations
Relative and absolute error thresholds
Error function verbosity
Linear solver type

Usage Considerations¶

Initial Guess: The quality of the initial guess can significantly affect the convergence and performance of the Gauss-Newton optimizer.
Non-convexity: Since the method relies on linear approximations, it may struggle with highly non-convex problems or those with poor initial estimates.
Performance: The Gauss-Newton method is generally faster than other nonlinear optimization methods like Levenberg-Marquardt for problems that are well-approximated by a quadratic model near the solution.

Files¶