# NAME

PDL::Opt::Simplex::Simple - A simplex optimizer for the rest of us (who may not know PDL).

# SYNOPSIS

```
use PDL::Opt::Simplex::Simple;
# Simple single-variable invocation
$simpl = PDL::Opt::Simplex::Simple->new(
vars => {
# initial guess for x
x => 1
},
f => sub {
my $vars = shift;
# Parabola with minima at x = -3
return (($vars->{x}+3)**2 - 5)
}
);
$simpl->optimize();
$result_vars = $simpl->get_result_simple();
print "x=" . $result_vars->{x} . "\n"; # x=-3
# Multi-vector Optimization and other settings:
$simpl = PDL::Opt::Simplex::Simple->new(
vars => {
# initial guess for arbitrarily-named vectors:
vec1 => { values => [ 1, 2, 3 ], enabled => [1, 1, 0] }
vec2 => { values => [ 4, 5 ], enabled => [0, 1] }
},
f => sub {
my ($vec1, $vec2) = ($_->{vec1}, $_->{vec2});
# do something with $vec1 and $vec2
# and return() the result to be minimized by simplex.
},
log => sub { }, # log callback
max_iter => 100, # max iterations
workers => 5, # parallelize with 5 forked workers (optional)
# simplex-specific options:
opts => {
# initial simplex size, smaller means less perturbation
ssize => 0.1,
},
);
$result_vars = $simpl->optimize();
use Data::Dumper;
print Dumper($result_vars);
```

# DESCRIPTION

This class uses PDL::Opt::Simplex to find the values for `vars`

that cause the `f`

coderef to return the minimum value. The difference between PDL::Opt::Simplex and PDL::Opt::Simplex::Simple is that PDL::Opt::Simplex expects all data to be in PDL format and it is more complicated to manage, whereas, PDL::Opt::Simplex:Simple uses all scalar Perl values. (PDL values are supported, too, see the PDL use case note below.)

With the original PDL::Opt::Simplex module, a single vector array had to be sliced into the different variables represented by the array. This was non-intuitive and error-prone. This class attempts to improve on that by defining data structure of variables, values, and whether or not a value is enabled for optimization.

This means you can selectively disable a particular value and it will be excluded from optimization but still included when passed to the user's callback function `f`

. Internal functions in this class compile the state of this variable structure into the vector array needed by simplex, and then extract values into a usable format to be passed to the user's callback function.

# FUNCTIONS

$self->new(%args) - Instantiate class

$self->optimize() - Run the optimization

$self->get_vars_expanded() - Returns the original

`vars`

in a fully expanded format$self->get_vars_simple() - Returns

`vars`

in the simplified formatThis format is suitable for passing into your

`f`

callback.$self->get_vars_orig() - Returns

`vars`

in originally passed format$self->get_result_expanded() - Returns the optimization result in expanded format.

$self->get_result_simple() - Returns the optimization result in the simplified format

This format is suitable for passing into your

`f`

callback.$self->set_vars(\%vars) - Set

`vars`

as if passed to the constructor.This can be used to feed a result from $self->get_result_expanded() into a new refined simplex iteration.

$self->set_ssize($ssize) - Set

`ssize`

as if passed to the constructor.Useful for calling simplex again with refined values

$self->scale_ssize($scale) - Multiply the current

`ssize`

by`$scale`

dumpify($vars)

This is for debugging:

Builds a tree from

`$vars`

that is suitable for passing to Data::Dumper. This is necessary because PDL's need to be stringified since Dumper() will dump at the object itself.

# ARGUMENTS

## * `vars`

- Hash of variables to optimize: the answer to your question.

### - Simple `vars`

Format

Thes are the variables being optimized to find a minimized result. The simplex() function returns minimized set of `vars`

. In its Simple Format, the `vars`

setting can assign values for vars directly as in the synopsis above:

```
vars => {
# initial guesses:
x => 1,
y => 2, ...
}
```

or as vectors of (possibly) different lengths:

```
vars => {
# initial guess for x
u => [ 4, 5, 6 ],
v => [ 7, 8 ], ...
}
```

### - Expanded `vars`

Format

You may find during optimization that it would be convenient to disable certain elements of the vector being optimized if, for example, you know that one value is already optimal but that it needs to be available to the f() callback. The expanded format shows that the 4th element is excluded from optimization by setting enabled=0 for that index.

Expanded format:

```
vars => {
varname => {
"values" => [...],
"minmax" => [ [min=>max], ...
"perturb_scale" => [...],
"enabled" => [...],
}, ...
}
```

`varname`

: the name of the variable being used.`values`

: an arrayref of values to be optimized`minmax`

: a double-array of min-max pairs (per index for vectors)-
Min-max pairs are clamped before being evaluated by simplex.

Note: for internal PDL::Opt::Simplex reasons, the

`max`

value is increased as`$var_max*1.0001`

, or one ten-thousandth bigger than the max value you request. See the comment at the top of`PDL::Opt::Simplex::Simple->_simplex_f()`

for details `round_result`

: Round the value to the nearest increment of this value upon completion-
You may need to round the final output values to a real-world limit after optimization is complete. Setting round_result will round after optimization finishes, but leave full precision while iterating. See also:

`round_each`

.This function uses Math::Round's

`nearest`

function:`nearest(10, 44) yields 40 nearest(10, 46) 50 nearest(10, 45) 50 nearest(25, 328) 325 nearest(.1, 4.567) 4.6 nearest(10, -45) -50`

`round_each`

: Round the value to the nearest increment of this value on each iteration.-
It is probably best to round at the end (

`round_result`

) to keep precision during each iteration, but the option is available in case you wish to use it. `perturb_scale`

: Scale parameter before being evaluated by simplex (per index for vectors)-
This is useful because Simplex's

`ssize`

parameter is the same for all values and you may find that some values need to be perturbed more or less than others while simulating. User interaction with`f`

and the result of`optimize`

will use the normally scaled values supplied by the user, this is just an internal scale for simplex.Internal details: The value passed to simplex is divided by perturb_scale parameter before being passed and multiplied by perturb_scale when returned. Thus, perturb_scale=0.1 would make simplex see the value as being 10x larger effectively perturbing it less, whereas, perturb_scale=10 would make it 10x smaller and perturb it more.

`enabled`

: 1 or 0: enabled a specific index to be optimized (per index for vectors)-
If 'enabled' is undefined then all values are enabled.

If 'enabled' is not an array, it can be a scalar 0 or 1 to indicate that all values are enabled/disabled. In this case your original structure will be replaced with an arrayref of all 0/1 values.

Enabling or disabling a variable may be useful in testing certain geometry charactaristics during optimization.

Internally, all values are vectors, even if the vectors are of length 1, but you can pass them as singletons like

`spaces`

as shorthand shown below instead of writing "spaces => [5]". In that example you can see that`spaces`

is disabled as well, so simplex will not optimize that value.`spaces => [ 5 ] # Element lengths vars => { lengths => { values => [ 1.038, 0.955, 0.959 ], minmax => [ [0.5=>1.5], [0.3=>1.2], [0.2=>1.1] ], perturb_scale => [ 10, 100, 1 ], enabled => [ 1, 1, 1 ], }, spaces => { values => 5, enabled => 0 }, ... }`

## * `f`

- Callback function to operate upon `vars`

The `f`

argument is a coderef that is called by the optimizer. It is passed a hashref of `vars`

in the Simple Format and must return a scalar result:

` f->({ lengths => [ 1.038, 0.955, 0.959, 0.949, 0.935 ], spaces => 5 });`

Note that a single-length vector will always be passed as a scalar to `f`

:

` vars => { x => [5] } will be passed as f->({ x => 5 })`

The Simplex algorithm will work to minimize the return value of your `f`

coderef, so return smaller values as your variables change to produce a (more) desired outcome.

## * `log`

- Callback function log status for each iteration.

```
log => sub {
my ($vars, $state) = @_;
print "LOG: " . Dumper($vars, $state);
}
```

The log() function is passed the current state of `vars`

in the same format as the `f`

callback. A second `$state`

argument is passed with information about the The return value is ignored. The following values are available in the `$state`

hashref:

```
{
'ssize' => '704.187123721893', # current ssize during iteration
'minima' => '53.2690700664067', # current minima returned by f()
'elapsed' => '3.12', # elapsed time in seconds since last log() call.
'srand' => 55294712, # the random seed for this run
'log_count' => 5, # how many times _log has been called
'optimization_pass' => 3, # pass# if multiple ssizes are used
'num_passes' => 6, # total number of passes
'best_pass' => 3, # the pass# that had the best goal result
'best_minima' => 0.2345 # The least value so far, returned by "f"
'best_vars' => { x=>1, ...} # The vars associated with "best_minima"
'log_count' => 22, # number of times log has been called in this pass
'iter_count' => 123, # number of f() has been called (including cache hits)
'prev_minima_count' => 10, # number of same minima's in a row
'cancel' => 0, # true if the simplex iteration is being cancelled
'all_vars' => [{x=>1},...], # multiple var options from simplex are logged here
'cache_hits' => 100, # Number of times simplex asked to try the same vars
'cache_misses' => 1000, # Number of times simplex asked to try unique vars
}
```

## * `ssize`

- Initial simplex size, see PDL::Opt::Simplex

Think of this as "step size" but not really, a bigger value makes larger jumps but the value doesn't translate to a unit. (It actually stands for simplex size, and it initializes the size of the simplex tetrahedron.)

You will need to scale the `ssize`

argument depending on your search space. Smaller `ssize`

values will search a smaller space of possible values provided in `vars`

. This is problem-space dependent and may require some trial and error to tune it where you need it to be.

Example for optimizing geometry in an EM simulation: Because it is proportional to wavelength, lower frequencies need a larger value and higher frequencies need a lower value.

The `ssize`

parameter may be an arrayref: If an arrayref is specified then it will run simplex to completion using the first ssize and then restart with the next `ssize`

value in the array. Each iteration uses the best result as the input to the next simplex iteration in an attempt to find increasingly better results. For example, 4 iterations with each `ssize`

one-half of the previous:

Note: `ssize`

is a simplex-specific option that must be placed in `{opts}`

:

```
opts => {
ssize => [ 4, 2, 1, 0.5 ]
}
```

Default: 1

## * `nocache`

- Disable result caching

By default we try not to re-calculate the same values. This is particularly useful when `round_each`

is used because it will round values from before passing them to `f`

, which increases the chance of a cache hit.

If you wish to disable caching then set "nocache => 1"

Default: undef (cache enabled)

## * `workers`

- Fork a number of workers to parallelize the computation

Some computations may benefit from parallization. Internally we use Parallel::Pipes to serialize `vars`

and send them to a worker process.

Example:

` workers => 5`

Default: undef (no forking)

## * `max_iter`

- Maximim number of Simplex iterations

Note that one Simplex iteration may call `f`

multiple times.

Default: 1000

## * `tolerance`

- Conversion tolerance for Simplex

The default is 1e-6. It tells Simplex to stop before `max_iter`

if very little change is being made between iterations.

Note: `tolerance`

is a simplex-specific option that must be placed in `{opts}`

:

```
opts => {
tolerance => 1e-3
}
```

Default: 1e-6

## * `srand`

- Value to seed srand

Simplex makes use of random perturbation, so setting this value will make the simulation deterministic from run to run.

The default when not defined is to call srand() without arguments and use a randomly generated seed. If set, it will call srand($self->{srand}) to initialize the initial seed. The result of this seed (whether passed or generated) is available in the status structure defined above.

Default: system generated.

## * `stagnant_minima_count`

- Abort the simplex iteration if the minima is not changing

This is the maximum number of iterations that can return a worse minima than the previous minima. Once reaching this limit the current iteration is cancelled due to stagnation. Setting this too low will provide poor results, setting it too high will just take longer to iterate when it gets stuck.

Note: This value may be somewhat dependent on the number of variables you are optimizing. The more variables, the bigger the value. A value of 30 seems to work well for 10 variables, so adjust if necessary.

Simplex will not cancel due to stagnation when `stagnant_minima_count`

is undefined.

Default: undef

## * `stagnant_minima_tolerance`

- threshold to count toward `stagnant_minima_count`

When `abs($prev_minima - $cur_minima) < $stagnant_minima_count`

then the iteration will be counted toward stagnation when `stagnant_minima_count`

is defined (see above). Otherwise, we assume progress is being made and the stagnation count is reset.

Default: same as `tolerance`

(see above)

`reduce_search`

- Reduce the search space

Sometimes PDL provides multiple variable sets to calculate during an iteration. If `reduce_search => 1`

is flagged then treat all variable sets as the same by only evaluating the first variable set and returning that result for all sets. This speeds up the optimization but may provide sub-optimal results.

This was the original behavior in back in Version 1.1, so newer versions are (probably) more accurate but will take longer to complete. However, it is still useful if you have a slow computation (`f`

) and want to converge sooner for an initial first pass. It is still recommended to run a final pass without `reduce_search`

.

Note: `reduce_search`

is a simplex-specific option that must be placed in `{opts}`

:

```
opts => {
reduce_search => 1,
...
}
```

# BEST PRACTICES AND USE CASES

## Antenna Geometry: Use an array for the `ssize`

parameter from coarse to fine perturbation.

This `PDL::Opt::Simplex::Simple`

module was originally written to optimize antenna geometries in conjunction with the "Optimizer Output" feature of the xnec2c (https://www.xnec2c.org) antenna simulator. The behavior is best described by Neoklis Kyriazis, 5B4AZ who originally wrote xnec2c: http://www.5b4az.org/pages/antenna_designs.html

```
"Xnec2c monitors its .nec input file for changes and re-runs the
frequency stepping loop which recalculates new data and prints to the
.csv file. It is therefore possible to arrange the optimizer program to
read the .csv data file, recalculate antenna parameters and save them
to the .nec input file.
Xnec2c will then recalculate and save new frequency-dependent data to
the .csv file. If the optimizer program is arranged to monitor changes
to the .csv file, then a continuous loop can be created in which new
antenna parameters are calculated and saved to the .nec file, new
frequency dependent data are calculated and saved to the .csv file and
the loop repeated until the desired results (optimization) are
obtained."
```

We find that a coarse "first pass" value for `ssize`

may not produce optimal results, so `PDL::Opt::Simplex::Simple`

will perform additional simplex iterations if you specify `ssize`

with multiple values to retry once a previous iteration finds a "good" (but not "great") result; the best minima from across all simplex passes is kept as the final result in case latter passes do not perform as well:

```
opts => {
ssize => [ 0.090, 0.075, 0.050, 0.025, 0.012 ]
}
```

This allows us to optimize antenna gain from 10.2 dBi with a single pass to 11.3 dBi after 5 passes, in addition to a much improved VSWR value.

See https://github.com/KJ7LNW/xnec2c-optimize for sample graphs and more information, including documentation to setup the demo so you can see `PDL::Opt::Simplex::Simple`

in action as the graphs update in real-time during the optimization process.

## PID Controller: Set ssize to 1 and scale perturb_scale for each variable.

We were using a proportional-integral-derivative ("PID") controller to optimize antenna motion for tracking orbiting satellites like the International Space Station. The goal is to minimize rotor overshoot and increase accuracy for the azimuth and elevation axis. Without getting into the PID controller implementation, just know that there are 3 primary terms in a PID controller that define its behavior (Kp, Ki, and Kd), and the satellite tracking is "good" if the overshoot is minimal. Here is a trivial implementation:

```
$simpl = PDL::Opt::Simplex::Simple->new(
vars => {
# initial guess for kp, ki, kd:
kp => 150,
ki => 120,
kd => 5
},
opts => {
ssize => 1,
},
f => sub {
my $vars = shift;
return track_satellite_get_overshoot(
kp => $vars->{kp},
ki => $vars->{ki},
kd => $vars->{kd});
}
);
print Dumper $simpl->optimize();
```

Note that `ssize=1`

so simplex will purturb the kp/ki/kd values in the range of about 1. This is great if you are already close to a solution, but in our case kp, ki, and kd need perturbed different amounts. It turns out that kd is quite small, while the optimal kp and ki values need a larger search space.

You might consider increasing `ssize`

, to `ssize=20`

but then kd will scale too quickly. To achieve this we used the extended variable format as follows:

```
$simpl = PDL::Opt::Simplex::Simple->new(
vars => {
# initial guess for kp, ki, kd:
kp => {
values => 150,
perturb_scale => 20,
},
ki => {
values => 120,
perturb_scale => 15,
},
kd => {
values => 5,
perturb_scale => 1,
},
},
opts => {
ssize => 1, # <- ssize is still set to 1 !
},
f => sub {
my $vars = shift;
return track_satellite_get_overshoot(
kp => $vars->{kp},
ki => $vars->{ki},
kd => $vars->{kd});
}
);
print Dumper $simpl->optimize();
```

As you can see above, the `perturb_scale`

value is different for each value; you could think of `perturb_scale`

as a "local ssize". Note that `ssize`

will still scale everything so if you wish to leave the relative scales defiend by `perturb_scale`

but double the search space, then set `ssize=2`

.

Ultimately simplex found the values to work best around Kp=190.90, Ki=166.33, and Kd=1.02. These values are specific to our hardware implementation (rotational mass, motor speed, etc) so the procedure is what is important here, not the values. Typically simplex is used against mathematical models, and it was interesting to run simplex against a real physical machine to calculate ideal values for its control.

If you are interested, here is a video about the antenna construction: https://youtu.be/Ab_oJHlENwo

## PDL variable considerations

You can use pdl's as vars in your code, but at the moment those pdl's must be singletons.

This will work:

```
->new({
vars => { x => pdl(5) }
}, ...)
```

but this will not:

```
->new({
vars => { x => pdl([1,2,3]) }
}, ...)
```

If you need PDL vectors in your `f()`

call then this could work because PDL::Opt::Simplex::Simple can optimize perl ARRAY ref's:

```
->new({
vars => { x => [1,2,3] }
},
f => sub {
my $vars = shift;
my $x = pdl $vars->{x};
# do stuff here, maybe return the sum:
return unpdl(sum $x);
},
...)
```

Future support for this is possible, but there is one major consideration: PDLs need to be generically decomposed into a 1-dimensionaly PDL before passing it to simplex() and then convert it back to the original N-dimensional form before passing it to the user's `f()`

call. This would then enable hash-named N-dimensional pdl optimization.

Patches welcome ;)

# ENVIRONMENT VARIABLES

`PDL_OPT_SIMPLEX_SIMPLE_WORKERS=N`

You can set

`PDL_OPT_SIMPLEX_SIMPLE_WORKERS`

in your environment to control the number of workers.Note: This environment variable overrides

`workers`

in`new()`

# SEE ALSO

## Upstream modules:

- PDL Implementation of Simplex: PDL::Opt::Simplex, http://pdl.perl.org/
- This modules github repository: https://github.com/KJ7LNW/perl-PDL-Opt-Simplex-Simple

## References:

- Wikipedia Article: https://en.wikipedia.org/wiki/Simplex_algorithm
- Video about how optimization algorithms like Simplex work, visually: https://youtu.be/NI3WllrvWoc

## Examples:

- Antenna Geometry Optimization: https://github.com/KJ7LNW/xnec2c-optimize
- PID Controller Optimization: https://github.com/KJ7NLL/space-ham/blob/master/optimize-pid.pl

## Other Optimization Implementations:

- PDL::Opt::ParticleSwarm - A PDL implementation of Particle Swarm
- PDL::Opt::ParticleSwarm::Simple - Use names for Particle Swarm-optimized values

# AUTHOR

Originally written at eWheeler, Inc. dba Linux Global Eric Wheeler to optimize antenna geometry for the https://www.xnec2c.org project.

# COPYRIGHT

Copyright (C) 2022 eWheeler, Inc. https://www.linuxglobal.com/

This module is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version.

This module is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with this module. If not, see <http://www.gnu.org/licenses/>.