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https://github.com/emmt/algorithms

Various useful numerical algorithms which can be used almost standalone.
https://github.com/emmt/algorithms

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Various useful numerical algorithms which can be used almost standalone.

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README 

          
# Algorithms



This repository is for various useful numerical algorithms which can be

used almost standalone.



## COBYLA



Directory [`cobyla`](cobyla) contains a C version of Mike Powell's COBYLA

(for **C**onstrained **O**ptimization **BY** **L**inear **A**pproximations)

algorithm for minimizing a function of many variables.  The method is

*derivatives free* (only the function values are needed) and accounts for

constraints on the variables.  The algorithm is described in:



> M.J.D. Powell, "A direct search optimization method that models the

> objective and constraint functions by linear interpolation," in Advances

> in Optimization and Numerical Analysis Mathematics and Its Applications,

> vol. 275 (eds. Susana Gomez and Jean-Pierre Hennart), Kluwer Academic

> Publishers, pp. 51-67 (1994).



A [Yorick](http://github.com/LLNL/yorick/) interface is also provided in

directory [`cobyla/yorick`](./cobyla/yorick).



## NEWUOA



Directory [`newuoa`](newuoa) provides a C implementation of Mike Powell's

NEWUOA algorithm for minimizing a function of many variables.  The method

is *derivatives free* (only the function values are needed) and accounts

for bound constraints on the variables.  The algorithm is described in:



> M.J.D. Powell, "The NEWUOA software for unconstrained minimization

> without derivatives", in Large-Scale Nonlinear Optimization, editors

> G. Di Pillo and M. Roma, Springer (2006), pages 255-297.



NEWUOA builds a quadratic model of the objective function from much less

than `(N+1)(N+2)/2` values of the function (with `N` the number of

variables).  The recommended number of points for building the quadratic

model is `2*N+1`.  For smooth objective functions, NEWUOA is expected to be

more efficient than COBYLA (which exploits a more simple linear model but

implements arbitrary inequality constraints while NEWUOA is unconstrained).

If you have bound constraints, you may consider using BOBYQA instead.



A [Yorick](http://github.com/LLNL/yorick/) interface is also provided in

directory [`newuoa/yorick`](./newuoa/yorick).



## BOBYQA



Directory [`bobyqa`](bobyqa) provides a C implementation of Mike Powell's

BOBYQA (for **B**ound constrained **O**ptimization **BY** **Q**uadratic

**A**pproximations) algorithm for minimizing a function of many variables.

The method is *derivatives free* (only the function values are needed) and

accounts for bound constraints on the variables.  The algorithm is

described in:



> M.J.D. Powell, "The BOBYQA Algorithm for Bound Constrained Optimization

> Without Derivatives."  Technical report, Department of Applied

> Mathematics and Theoretical Physics, University of Cambridge (2009).



BOBYQA builds a quadratic model of the objective function from much less

than `(N+1)(N+2)/2` values of the function (with `N` the number of

variables).  The recommended number of points for building the quadratic

model is `2*N+1`.  For smooth objective functions, BOBYQA is expected to be

more efficient than COBYLA (which exploits a more simple linear model but

implements arbitrary inequality constraints).



## TOLMIN



Directory [`tolmin/src`](tolmin/src) provides a C implementation of Mike

Powell's TOLMIN algorithm for minimizing a function of many variables subject

to linear and bound constraints.  The algorithm is described in:



> M.J.D. Powell, "A tolerant algorithm for linearly constrained optimization

> calculations", Math. Programming B, Vol. 45, pp. 547-566 (1989).



The present code is based on the original FORTRAN version written by Mike

Powell who released his code under the GNU Lesser General Public License.  His

original code is available at

[CCPForge](https://ccpforge.cse.rl.ac.uk/gf/project/powell/).



## LINCOA



Directory [`lincoa/src`](lincoa/src) provides a C implementation of Mike

Powell's LINCOA algorithm for minimizing a function of many variables subject

to linear constraints.  The algorithm is described in:



> M.J.D. Powell, "On fast trust region methods for quadratic models with linear

> constraints", Report of the Department of Applied Mathematics and Theoretical

> Physics, Cambridge University, DAMTP 2014/NA02 (2014).



The present code is based on the original FORTRAN version written by Mike

Powell who released his code under the GNU Lesser General Public License.  His

original code is available at

[CCPForge](https://ccpforge.cse.rl.ac.uk/gf/project/powell/).



## L-BFGS-B



L-BFGS-B (for *Limited memory BFGS method with Bounds*) is an algorithm by

R.H. Byrd, P. Lu, J. Nocedal and C. Zhu (see references below) to minimize

a smooth function of many variables with simple bound constraints.  The

method requires the computation of the function and its gradient.  It exploits

a limited memory approximation of the function Hessian with BFGS updates.

It may be used to solve large scale problems.



Directory [`lbfgsb`](./lbfgsb) contains original code of L-BFGS-B with

simple C and Yorick wrapper code to use L-BFGS-B in these languages.



The L-BFGS-B algorithm is described in:



1. R.H. Byrd, P. Lu, J. Nocedal and C. Zhu, *A limited memory algorithm for

   bound constrained optimization*, SIAM J. Scientific Computing 16 (1995),

   no. 5, pp. 1190--1208.



2. C. Zhu, R.H. Byrd, P. Lu, J. Nocedal, *L-BFGS-B: a limited memory

   FORTRAN code for solving bound constrained optimization problems*,

   Tech. Report, NAM-11, EECS Department, Northwestern University, 1994.