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Algorithm::LBFGS - A Perl6 bindings for libLBFGS

```
use Algorithm::LBFGS;
use Algorithm::LBFGS::Parameter;
my Algorithm::LBFGS $lbfgs .= new;
my &evaluate = sub ($instance, $x, $g, $n, $step --> Num) {
my Num $fx = ($x[0] - 2.0) ** 2 + ($x[1] - 5.0) ** 2;
$g[0] = 2.0 * $x[0] - 4.0;
$g[1] = 2.0 * $x[1] - 10.0;
return $fx;
};
my Algorithm::LBFGS::Parameter $parameter .= new;
my Num @x0 = [0e0, 0e0];
my @x = $lbfgs.minimize(:@x0, :&evaluate, :$parameter);
@x.say; # [2e0, 5e0]
```

Algorithm::LBFGS is a Perl6 bindings for libLBFGS. libLBFGS is a C port of the implementation of Limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) method written by Jorge Nocedal.

The L-BFGS method solves the unconstrainted minimization problem,

```
minimize F(x), x = (x1, x2, ..., xN),
```

only if the objective function F(x) and its gradient G(x) are computable.

```
my $lbfgs = Algorithm::LBFGS.new;
my Algorithm::LBFGS $lbfgs .= new; # with type restrictions
```

```
my @x = $lbfgs.minimize(:@x0!, :&evaluate, :&progress, :$parameter); # use &progress callback
my @x = $lbfgs.minimize(:@x0!, :&evaluate, :$parameter);
```

Runs the optimization and returns the resulting variables.

`:@x0`

is the initial value of the variables.

`:&evaluate`

is the callback function. This requires the definition of the objective function F(x) and its gradient G(x).

`:&progress`

is the callback function. This gets called on every iteration and can output the internal state of the current iteration.

`:$parameter`

is the instance of the `Algorithm::LBFGS::Parameter`

class.

The one of the simplest `&evaluate`

callback function would be like the following:

```
my &evaluate = sub ($instance, $x, $g, $n, $step --> Num) {
my Num $fx = ($x[0] - 2.0) ** 2 + ($x[1] - 5.0) ** 2; # F(x) = (x0 - 2.0)^2 + (x1 - 5.0)^2
# G(x) = [∂F(x)/∂x0, ∂F(x)/∂x1]
$g[0] = 2.0 * $x[0] - 4.0; # ∂F(x)/∂x0 = 2.0 * x0 - 4.0
$g[1] = 2.0 * $x[1] - 10.0; # ∂F(x)/∂x1 = 2.0 * x1 - 10.0
return $fx;
};
```

`$instance`

is the user data. (NOTE: NYI in this binder. You must set it as a first argument, but you can't use it in the callback.)`$x`

is the current values of variables.`$g`

is the current gradient values of variables.`$n`

is the number of variables.`$step`

is the line-search step used for this iteration.

`&evaluate`

requires all of these five arguments in this order.

After writing the definition of the objective function F(x) and its gradient G(x), it requires returning the value of the F(x).

The one of the simplest `&progress`

callback function would be like the following:

```
my &progress = sub ($instance, $x, $g, $fx, $xnorm, $gnorm, $step, $n, $k, $ls --> Int) {
"Iteration $k".say;
"fx = $fx, x[0] = $x[0], x[1] = $x[1]".say;
return 0;
}
```

`$instance`

is the user data. (NOTE: NYI in this binder. You must set it as a first argument, but you can't use it in the callback.)`$x`

is the current values of variables.`$g`

is the current gradient values of variables.`$fx`

is the current value of the objective function.`$xnorm`

is the Euclidean norm of the variables.`$gnorm`

is the Euclidean norm of the gradients.`$step`

is the line-search step used for this iteration.`$n`

is the number of variables.`$k`

is the iteration count.`$ls`

the number of evaluations called for this iteration.

`&progress`

requires all of these ten arguments in this order.

Below is the examples of creating a

```
my Algorithm::LBFGS::Parameter $parameter .= new; # sets default parameter
my Algorithm::LBFGS::Parameter $parameter .= new(max_iterations => 100); # sets max_iterations => 100
```

Int

`m`

is the number of corrections to approximate the inverse hessian matrix.Num

`epsilon`

is epsilon for convergence test.Int

`past`

is the distance for delta-based convergence test.Num

`delta`

is delta for convergence test.Int

`max_iterations`

is the maximum number of iterations.Int

`linesearch`

is the line search algorithm. This requires one of`LBFGS_LINESEARCH_DEFAULT`

,`LBFGS_LINESEARCH_MORETHUENTE`

,`LBFGS_LINESEARCH_BACKTRACKING_ARMIJO`

,`LBFGS_LINESEARCH_BACKTRACKING`

,`LBFGS_LINESEARCH_BACKTRACKING_WOLFE`

and`LBFGS_LINESEARCH_BACKTRACKING_STRONG_WOLFE`

. The default value is`LBFGS_LINESEARCH_MORETHUENTE`

.Int

`max_linesearch`

is the maximum number of trials for the line search.Num

`min_step`

is the minimum step of the line search routine.Num

`max_step`

is the maximum step of the line search.Num

`ftol`

is a parameter to control the accuracy of the line search routine.Num

`wolfe`

is a coefficient for the Wolfe condition.Num

`gtol`

is a parameter to control the accuracy of the line search routine.Num

`xtol`

is the machine precision for floating-point values.Num

`orthantwise_c`

is a coeefficient for the L1 norm of variables.Int

`orthantwise_start`

is the start index for computing L1 norm of the variables.Int

`orthantwise_end`

is the end index for computing L1 norm of the variables.

TBD

titsuki titsuki@cpan.org

Copyright 2016 titsuki

Copyright 1990 Jorge Nocedal

Copyright 2007-2010 Naoki Okazaki

libLBFGS by Naoki Okazaki is licensed under the MIT License.

This library is free software; you can redistribute it and/or modify it under the terms of the MIT License.