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im kinda confused and searching for the right method i need to use.

I got a function $F:\mathbb{R}^3\rightarrow \mathbb{R}^3$ for which i need to solve $F(n_1,n_2,\theta)=0$ but with the constraint $||n||\le 1$.

I first forgot about the constraint, was using just the multivariate Newton-Raphson method but obviously thats not working. I found some informations on penalty methods and barrier-based interior-point algorithms like on https://en.wikipedia.org/wiki/Interior_point_method

The first problem is that there are always minimization problems, i think i can rewrite my problem to min $|F|$. The second problem is that in the methods which built in the constraints F is always $F:\mathbb{R^n}\rightarrow \mathbb{R}$ which is not my case. What do i do? What do i use?

José Carlos Santos
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korzi
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  • You could minimize $||F||^2$ to mantain differentiability ,then look for the KKT conditions. – Koto Sep 28 '17 at 12:00

1 Answers1

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This problem occurs frequently in computational physics. The main idea is to have a vector penalization method that remains on the positive half space (positive projection) of the residual function F. This penalty term acts as a barrier to steer the solution path away from prohibited regions of the solution space. I have one reference on a method that avoids forming a minimization problem: https://www.researchgate.net/publication/265963806_Constrained_pseudo-transient_continuation https://onlinelibrary.wiley.com/doi/abs/10.1002/nme.4858

Marco Ceze
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