'\"
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.TH "math::probopt" n 1\&.1 tcllib "Tcl Math Library"
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.SH NAME
math::probopt \- Probabilistic optimisation methods
.SH SYNOPSIS
package require \fBTcl 8\&.6 9\fR
.sp
package require \fBTclOO\fR
.sp
package require \fBmath::probopt 1\&.1\fR
.sp
\fB::math::probopt::pso\fR \fIfunction\fR \fIbounds\fR \fIargs\fR
.sp
\fB::math::probopt::sce\fR \fIfunction\fR \fIbounds\fR \fIargs\fR
.sp
\fB::math::probopt::diffev\fR \fIfunction\fR \fIbounds\fR \fIargs\fR
.sp
\fB::math::probopt::lipoMax\fR \fIfunction\fR \fIbounds\fR \fIargs\fR
.sp
\fB::math::probopt::adaLipoMax\fR \fIfunction\fR \fIbounds\fR \fIargs\fR
.sp
.BE
.SH DESCRIPTION
.PP
The purpose of the \fBmath::probopt\fR 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\&.
.PP
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, \fI-iterations 100\fR\&.
.PP
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\&.
.PP
A few notes, however:
.IP \(bu
With the exception of LIPO, the algorithms are capable of dealing with irregular (non-smooth) and even discontinuous
functions\&.
.IP \(bu
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\&.
.PP
.PP
The collection consists of the following algorithms:
.IP \(bu
PSO - particle swarm optimisation
.IP \(bu
SCE - shuffled complexes evolution
.IP \(bu
DE - differential evolution
.IP \(bu
LIPO - Lipschitz optimisation
.PP
The various procedures have a uniform interface:
.CS


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

.CE
The arguments have the following meaning:
.IP \(bu
The argument \fIfunction\fR is the name of the procedure that evaluates the function\&.
Its interface is:
.CS


    set value [function coords]

.CE
.IP
where \fIcoords\fR is a list of coordinates at which to evaluate the function\&. It is
supposed to return the function value\&.
.IP \(bu
The argument \fIbounds\fR 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 \fIx**2 + (y-1)**4\fR, you may specify
the bounds as \fI{{-1 1} {-1 1}}\fR, that is, two pairs for the two coordinates\&.
.IP \(bu
The rest (\fIargs\fR) consists of zero or more key-value pairs to specify the options\&. Which
options are supported by which algorithm, is documented below\&.
.PP
The result of the various optimisation procedures is a dictionary containing at least the
following elements:
.IP \(bu
\fIoptimum-coordinates\fR is a list containing the coordinates of the optimum that was found\&.
.IP \(bu
\fIoptimum-value\fR is the function value at those coordinates\&.
.IP \(bu
\fIevaluations\fR is the number of function evaluations\&.
.IP \(bu
\fIbest-values\fR is a list of successive best values, obtained as
part of the iterations\&.
.PP
.SH "DETAILS ON THE ALGORITHMS"
The algorithms in the package are the following:
.TP
\fB::math::probopt::pso\fR \fIfunction\fR \fIbounds\fR \fIargs\fR
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\&.
.sp
It recognises the following options:
.RS
.IP \(bu
\fI-swarmsize number\fR: Number of particles to consider (default: 50)
.IP \(bu
\fI-vweight    value\fR: Weight for the current "velocity" (0-1, default: 0\&.5)
.IP \(bu
\fI-pweight    value\fR: Weight for the individual particle's best position (0-1, default: 0\&.3)
.IP \(bu
\fI-gweight    value\fR: Weight for the "best" overall position as per particle (0-1, default: 0\&.3)
.IP \(bu
\fI-type       local/global\fR: Type of optimisation
.IP \(bu
\fI-neighbours number\fR: Size of the neighbourhood (default: 5, used if "local")
.IP \(bu
\fI-iterations number\fR: Maximum number of iterations
.IP \(bu
\fI-tolerance  value\fR: Absolute minimal improvement for minimum value
.RE
.TP
\fB::math::probopt::sce\fR \fIfunction\fR \fIbounds\fR \fIargs\fR
The "shuffled complex evolution" algorithm is an extension of the Nelder-Mead algorithm that
uses multiple complexes and reorganises these complexes to find the "global" optimum\&.
.sp
It recognises the following options:
.RS
.IP \(bu
\fI-complexes           number\fR: Number of particles to consider (default: 2)
.IP \(bu
\fI-mincomplexes        number\fR: Minimum number of complexes (default: 2; not currently used)
.IP \(bu
\fI-newpoints           number\fR: Number of new points to be generated (default: 1)
.IP \(bu
\fI-shuffle             number\fR: Number of iterations after which to reshuffle the complexes (if set to 0, the default, a number will be calculated from the number of dimensions)
.IP \(bu
\fI-pointspercomplex    number\fR: Number of points per complex (if set to 0, the default, a number will be calculated from the number of dimensions)
.IP \(bu
\fI-pointspersubcomplex number\fR: Number of points per subcomplex (used to select the best points in each complex; if set to 0, the default, a number will be calculated from the number of dimensions)
.IP \(bu
\fI-iterations          number\fR: Maximum number of iterations (default: 100)
.IP \(bu
\fI-maxevaluations      number\fR: Maximum number of function evaluations (when this number is reached the iteration is broken off\&. Default: 1000 million)
.IP \(bu
\fI-abstolerance        value\fR: Absolute minimal improvement for minimum value (default: 0\&.0)
.IP \(bu
\fI-reltolerance        value\fR: Relative minimal improvement for minimum value (default: 0\&.001)
.RE
.TP
\fB::math::probopt::diffev\fR \fIfunction\fR \fIbounds\fR \fIargs\fR
The "differential evolution" algorithm uses a number of initial points that are then updated using randomly selected points\&. It is more or less akin
to genetic algorithms\&. It is controlled by two parameters, factor and lambda, where the first determines the update via random points and the second
the update with the best point found sofar\&.
.sp
It recognises the following options:
.RS
.IP \(bu
\fI-iterations          number\fR: Maximum number of iterations (default: 100)
.IP \(bu
\fI-number              number\fR: Number of point to work with (if set to 0, the default, it is calculated from the number of dimensions)
.IP \(bu
\fI-factor              value\fR: Weight of randomly selected points in the updating (0-1, default: 0\&.6)
.IP \(bu
\fI-lambda              value\fR: Weight of the best point found so far in the updating (0-1, default: 0\&.0)
.IP \(bu
\fI-crossover           value\fR: Fraction of new points to be considered for replacing the old ones (0-1, default: 0\&.5)
.IP \(bu
\fI-maxevaluations      number\fR: Maximum number of function evaluations (when this number is reached the iteration is broken off\&. Default: 1000 million)
.IP \(bu
\fI-abstolerance        value\fR: Absolute minimal improvement for minimum value (default: 0\&.0)
.IP \(bu
\fI-reltolerance        value\fR: Relative minimal improvement for minimum value (default: 0\&.001)
.RE
.TP
\fB::math::probopt::lipoMax\fR \fIfunction\fR \fIbounds\fR \fIargs\fR
The "Lipschitz optimisation" algorithm uses the "Lipschitz" property of the given function to find a \fImaximum\fR in the given bounding box\&. There are
two variants, \fIlipoMax\fR assumes a fixed estimate for the Lipschitz parameter\&.
.sp
It recognises the following options:
.RS
.IP \(bu
\fI-iterations          number\fR: Number of iterations (equals the actual number of function evaluations, default: 100)
.IP \(bu
\fI-lipschitz           value\fR: Estimate of the Lipschitz parameter (default: 10\&.0)
.RE
.TP
\fB::math::probopt::adaLipoMax\fR \fIfunction\fR \fIbounds\fR \fIargs\fR
The "adaptive Lipschitz optimisation" algorithm uses the "Lipschitz" property of the given function to find a \fImaximum\fR in the given bounding box\&. The adaptive
variant actually uses two phases to find a suitable estimate for the Lipschitz parameter\&. This is controlled by the "Bernoulli" parameter\&.
.sp
When you specify a large number of iterations, the algorithm may take a very long time to complete as it is trying to improve on the Lipschitz parameter and
the chances of hitting a better estimate diminish fast\&.
.sp
It recognises the following options:
.RS
.IP \(bu
\fI-iterations          number\fR: Number of iterations (equals the actual number of function evaluations, default: 100)
.IP \(bu
\fI-bernoulli           value\fR: Parameter for random decisions (exploration versus exploitation, default: 0\&.1)
.RE
.PP
.SH REFERENCES
The various algorithms have been described in on-line publications\&. Here are a few:
.IP \(bu
\fIPSO\fR: Maurice Clerc, Standard Particle Swarm Optimisation (2012)
\fIhttps://hal\&.archives-ouvertes\&.fr/file/index/docid/764996/filename/SPSO_descriptions\&.pdf\fR
.sp
Alternatively: \fIhttps://en\&.wikipedia\&.org/wiki/Particle_swarm_optimization\fR
.IP \(bu
\fISCE\fR: Qingyuan Duan, Soroosh Sorooshian, Vijai K\&. Gupta, Optimal use offo the SCE-UA global optimization method for calibrating watershed models
(1994), Journal of Hydrology 158, pp 265-284
.sp
\fIhttps://www\&.researchgate\&.net/publication/223408756_Optimal_Use_of_the_SCE-UA_Global_Optimization_Method_for_Calibrating_Watershed_Models\fR
.IP \(bu
\fIDE\fR: Rainer Storn and Kenneth Price, Differential Evolution - A simple and efficient adaptivescheme for globaloptimization over continuous spaces
(1996)
.sp
\fIhttp://www1\&.icsi\&.berkeley\&.edu/~storn/TR-95-012\&.pdf\fR
.IP \(bu
\fILIPO\fR: Cedric Malherbe and Nicolas Vayatis, Global optimization of Lipschitz functions,
(june 2017)
.sp
\fIhttps://arxiv\&.org/pdf/1703\&.02628\&.pdf\fR
.PP
.SH KEYWORDS
mathematics, optimisation, probabilistic calculations
.SH CATEGORY
Mathematics