probopt - Man Page

Probabilistic optimisation methods

Synopsis

package require Tcl 8.6

package require TclOO

package require math::probopt 1

::math::probopt::pso function bounds args

::math::probopt::sce function bounds args

::math::probopt::diffev function bounds args

::math::probopt::lipoMax function bounds args

::math::probopt::adaLipoMax function bounds args

The purpose of the math::probopt package is to provide various optimisation algorithms that are based on probabilistic techniques. The results of these algorithms may therefore vary from one run to the next. The algorithms are all well-known and well described and proponents generally claim they are efficient and reliable.

As most of these algorithms have one or more tunable parameters or even variations, the interface to each accepts options to set these parameters or the select the variation. These take the form of key-value pairs, for instance, -iterations 100.

This manual does not offer any recommendations with regards to these algorithms, nor does it provide much in the way of guidelines for the parameters. For this we refer to online articles on the algorithms in question.

A few notes, however:

With the exception of LIPO, the algorithms are capable of dealing with irregular (non-smooth) and even discontinuous functions.
The results depend on the random number seeding and are likely not to be very accurate, especially if the function varies slowly in the vicinty of the optimum. They do give a good starting point for a deterministic algorithm.

The collection consists of the following algorithms:

PSO - particle swarm optimisation
SCE - shuffled complexes evolution
DE - differential evolution
LIPO - Lipschitz optimisation

The various procedures have a uniform interface:

   set result [::math::probopt::algorithm function bounds args]

The arguments have the following meaning:

The argument function is the name of the procedure that evaluates the function. Its interface is:
```
    set value [function coords]
```
where coords is a list of coordinates at which to evaluate the function. It is supposed to return the function value.
The argument bounds is a list of pairs of minimum and maximum for each coordinate. This list implicitly determines the dimension of the coordinate space in which the optimum is to be sought, for instance for a function like x**2 + (y-1)**4, you may specify the bounds as {{-1 1} {-1 1}}, that is, two pairs for the two coordinates.
The rest (args) consists of zero or more key-value pairs to specify the options. Which options are supported by which algorithm, is documented below.

The result of the various optimisation procedures is a dictionary containing at least the following elements:

optimum-coordinates is a list containing the coordinates of the optimum that was found.
optimum-value is the function value at those coordinates.
evaluations is the number of function evaluations.
best-values is a list of successive best values, obtained as part of the iterations.

Details on the Algorithms

The algorithms in the package are the following:

::math::probopt::pso function bounds args

The "particle swarm optimisation" algorithm uses the idea that the candidate optimum points should swarm around the best point found so far, with variations to allow for improvements.