How to regularize the Levenberg-Marquardt algorithm in lsqnonlin?

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Abhilash Awasthi
Abhilash Awasthi 2022-7-27
编辑： Matt J 2022-7-27
I need to solve a non-linear least square problem using Levenberg-Marquardt (LM) method but with some additional terms in the objective function (see below)
lsqnonlin function in MATLAB optimization toolbox only takes ( as input and updates the parameters using the normal equations for the least-square problem [1]. Is there any way to solve such problems using lsqnonlin?
[1] Moré, J.J., 1978. The Levenberg-Marquardt algorithm: implementation and theory. In Numerical analysis (pp. 105-116). Springer, Berlin, Heidelberg.

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回答（1 个）

Matt J
Matt J 2022-7-27
编辑：Matt J 2022-7-27
You can append the square root of the regularization term to the vector . However, your objective function does not look differentiable, which Levenberg-Marquardt assumes, so you might have to use a differentiable approximation to the TV term.
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Torsten
Torsten 2022-7-27
编辑：Torsten 2022-7-27
And in the end, PI is one scalar value - some sort of cumulated approximation error over complete Omega ?
In this case, fmincon is better suited to solve your problem compared to lsqnonlin, I guess, because PI already performs summation of the errors squared in the discretization points whereas lsqnonlin would work with the errors in the discretization points separately.
Matt J
Matt J 2022-7-27
编辑：Matt J 2022-7-27
@Abhilash Awasthi If you construct the vector-valued function,
F(theta)=[u(theta)-um;
sqrt(2*regularization(theta))]
and give this F(theta) to lsqnonlin as the objective function, then lsqnonlin will minimize the regularized objective function H(theta) that you have posted.

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