Is there any way to define a loss function in optimization problems?

Is there any way to define a loss function when setting up an optimization problem using the optimization toolbox?

Main purpose: outlier detection.

Examples for loss functions in other optimization libraries:

• Ceres solver: http://ceres-solver.org/nnls_modeling.html?highlight=loss#lossfunction
• scipy.optimize.least_squares: https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.least_squares.html

Answer 1

[Matt J]

Yes, absolutely. Without a function to optimize, it's not an optimization problem.

[Philipp Glira]

My questions was regarding loss functions.

The main purpose of a loss function is the detection of outliers. In other words, it makes the optimization robust against gross observation errors (=outliers).

I don't see any possibility in the documentation of the optimization toolbox to define such a loss function, but maybe I have just overlooked that part.

To clarify:

1) non-robust optimization, i.e. without loss function:

(see the 3 outliers with large y values)

2) robust optimization, i.e. with loss function:

(the 3 outliers are detected as outliers and thus have no influence on the estimated unknowns)

[Matt J]

No, there are no outlier rejection utilities in the Optimization Toolbox solvers. The Computer Vision Toolbox, however, does have a RANSAC routine,

https://www.mathworks.com/help/vision/ref/ransac.html

You could also try removing outliers with rmoutliers,

https://www.mathworks.com/help/matlab/ref/rmoutliers.html