Algorithm for multidimensional constrained root finding

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SUMMARY

The discussion focuses on finding a suitable algorithm for multidimensional constrained root finding, specifically for a dynamic model with 60 state variables and coupled differential equations. The user is currently using an adapted Newton-Raphson method from Numerical Recipes, which lacks support for bounds and often converges to invalid values. They are considering the BFGS algorithm, which supports constraints, but are uncertain about its applicability for root finding. The consensus is to implement and test the BFGS algorithm to evaluate its effectiveness.

PREREQUISITES
  • Understanding of multidimensional root finding techniques
  • Familiarity with Fortran programming language
  • Knowledge of the BFGS algorithm and its applications
  • Experience with numerical methods, specifically Newton-Raphson and constrained optimization
NEXT STEPS
  • Implement the BFGS algorithm for constrained root finding in Fortran
  • Research the differences between minimization and root finding algorithms
  • Explore alternative algorithms for constrained root finding, such as the Augmented Lagrange method
  • Study the implementation of the L-BFGS-B algorithm for large-scale problems
USEFUL FOR

Researchers and developers working on dynamic models, numerical analysts, and anyone involved in solving multidimensional constrained optimization problems.

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Hi all,

I'm looking for an algorithm for multidimensional constrained root finding, implemented in Fortran. It's intended for finding a steady-state solution for a dynamic model. I have n state variables and n coupled differential equations (n~=60), and I need to find the value for the state variables at which the rate of change is zero.

Currently I'm working with an adapted version of Newton-Raphson taken from Numerical Recipes, but this algorithm doesn't support bounds and the solver has a tendency to converge on impossible values. Actually, it seems that not many multidimensional algorithms support bounds. I found an implementation of BFGS (http://hod.greeley.org/papers/Unsorted/lbfgsb.pdf) that supports constraints, but I'm not sure if this suitable for my purpose. From what I understand, not every minimization algorithm is suitable for root finding.
Can anyone tell me if the BGFS algorithm is suitable for root finding, or suggest a better algorithm?

Many thanks in advance!
 
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It is probably best to implement it and test it. Should be faster than searching for other algorithms without knowing the outcome.
 

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