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Aide Scilab >> Optimisation et Simulation > Optimization simplex > optimsimplex

optimsimplex

Manage a simplex with arbitrary number of points.

SYNOPSIS

[ newobj [, data] ] = optimsimplex_new ( coords , fun [, data] )
[ newobj [, data] ] = optimsimplex_new ( "axes" , x0 , fun , len [, data] )
[ newobj [, data] ] = optimsimplex_new ( "spendley" , x0 , fun , len [, data] )
[ newobj [, data] ] = optimsimplex_new ( "pfeffer" , x0 , fun , deltausual , deltazero [, data] )
[ newobj [, data] ] = optimsimplex_new ( "randbounds" , x0 , fun , boundsmin , boundsmax , nbve  [, data] )
[ newobj [, data] ] = optimsimplex_new ( "oriented" , simplex0 , fun [, data] )
this = optimsimplex_destroy (this)
this = optimsimplex_setall ( this , simplex )
this = optimsimplex_setallfv ( this , fv )
this = optimsimplex_setallx ( this , x )
this = optimsimplex_setfv ( this , ive , fv )
this = optimsimplex_setn ( this , n )
this = optimsimplex_setnbve ( this , nbve )
this = optimsimplex_setve ( this , ive , fv , x )
this = optimsimplex_setx ( this , ive , x )
simplex = optimsimplex_getall ( this )
fv = optimsimplex_getallfv ( this )
x = optimsimplex_getallx ( this )
fv = optimsimplex_getfv ( this , ive )
n = optimsimplex_getn ( this )
nbve = optimsimplex_getnbve ( this )
vertex = optimsimplex_getve ( this , ive )
x = optimsimplex_getx ( this , ive )
sicenter = optimsimplex_center ( this )
optimsimplex_check ( this )
[ this , data ] = optimsimplex_computefv ( this , fun , data )
df = optimsimplex_deltafv ( this )
dfm = optimsimplex_deltafvmax ( this )
m = optimsimplex_dirmat ( this )
m = optimsimplex_fvmean ( this )
sd = optimsimplex_fvstdev ( this )
v = optimsimplex_fvvariance ( this )
[ g , data ] = optimsimplex_gradientfv ( this , fun , method , data )
[ r , data ] = optimsimplex_reflect ( this , fun , data )
[ this , data ] = optimsimplex_shrink ( this , fun , sigma , data )
ssize = optimsimplex_size ( this , method )
this = optimsimplex_sort ( this )
str = optimsimplex_tostring ( this )
cen = optimsimplex_xbar ( this , iexcl )

Purpose

The goal of this component is to provide a building block for optimization algorithms based on a simplex. The optimsimplex package may be used in the following optimization methods :

  • the Spendley et al. simplex method,

  • the Nelder-Mead method,

  • the Box algorithm for constrained optimization,

  • the multi-dimensional search by Virginia Torczon,

  • etc ...

Design

This toolbox is designed with Oriented Object ideas in mind.

Features

The following is a list of features the Nelder-Mead prototype algorithm currently provides :

  • Manage various simplex initializations

    • initial simplex given by user,

    • initial simplex computed with a length and along the coordinate axes,

    • initial regular simplex computed with Spendley et al. formula,

    • initial simplex computed by a small perturbation around the initial guess point,

    • initial simplex computed from randomized bounds.

  • sort the vertices by increasing function values,

  • compute the standard deviation of the function values in the simplex,

  • compute the simplex gradient with forward or centered differences,

  • shrink the simplex toward the best vertex,

  • etc...

Description

This set of commands allows to manage a simplex made of k>=n+1 points in a n-dimensional space. This component is the building block for a class of direct search optimization methods such as the Nelder-Mead algorithm or Torczon's Multi-Dimensionnal Search.

A simplex is designed as a collection of k>=n+1 vertices. Each vertex is made of a point and a function value at that point.

The simplex can be created with various shapes. It can be configured and quieried at will. The simplex can also be reflected or shrinked. The simplex gradient can be computed with a order 1 forward formula and with a order 2 centered formula.

The optimsimplex_new function allows to create a simplex. If vertices coordinates are given, there are registered in the simplex. If a function is provided, it is evaluated at each vertex. The optimsimplex_destroy function destroys the object and frees internal memory. Several functions allow to create a simplex with special shapes, including axes-by-axes (optimsimplex_axes), regular (optimsimplex_spendley), randomized bounds simplex with arbitrary nbve vertices (optimsimplex_randbounds) and an heuristical small variation around a given point (optimsimplex_pfeffer).

In the following functions, simplices and vertices are, depending on the functions either input or output arguments. The following general principle have been used to manage the storing of the coordinates of the points.

  • The vertices are stored row by row, while the coordinates are stored column by column. This implies the following rules.

  • The coordinates of a vertex are stored in a row vector, i.e. a 1 x n matrix where n is the dimension of the space.

  • The function values are stored in a column vector, i.e. a nbve x 1 matrix where nbve is the number of vertices.

Functions

The following functions are available.

newobj = optimsimplex_new ( )

Creates a new simplex object. All input arguments are optional. If no input argument is provided, returns an empty simplex object.

The following is a complete list of available calling sequences.

newobj = optimsimplex_new ( coords )
newobj = optimsimplex_new ( coords , fun )
[ newobj , data ] = optimsimplex_new ( coords , fun , data )
newobj

The new object.

coords

optional, matrix of point coordinates in the simplex.

The coords matrix is expected to be a nbve x n matrix, where n is the dimension of the space and nbve is the number of vertices in the simplex, with nbve>= n+1.

fun

optional, the function to compute at vertices.

The function is expected to have the following input and output arguments :

function y=myfunction(x)

where x is a row vector.

data

optional, user-defined data passed to the function.

If data is provided, it is passed to the callback function both as an input and output argument. In that case, the function must have the following header :

function [y, data]=myfunction(x, data)

The data input parameter may be used if the function uses some additionnal parameters. It is returned as an output parameter because the function may modify the data while computing the function value. This feature may be used, for example, to count the number of times that the function has been called.

this = optimsimplex_new ( "axes" , x0 )

Creates a new simplex object so that it is computed axis by axis, with the given length.

The following is a complete list of available calling sequences.

this = optimsimplex_new ( "axes" , x0 , fun )
this = optimsimplex_new ( "axes" , x0 , fun , len )
[ this , data ] = optimsimplex_new ( "axes" , x0 , fun , len , data )
this

The current simplex object.

x0

the initial point, as a row vector.

fun

optional, the function to compute at vertices.

The function is expected to have the following input and output arguments :

function y=myfunction(x)
len

optional, the length of the simplex. The default length is 1.0. If length is a value, that unique length is used in all directions. If length is a vector with n values, each length is used with the corresponding direction.

data

optional, user-defined data passed to the function.

If data is provided, it is passed to the callback function both as an input and output argument. In that case, the function must have the following header :

function [y, data]=myfunction(x, data)

The data input parameter may be used if the function uses some additionnal parameters. It is returned as an output parameter because the function may modify the data while computing the function value. This feature may be used, for example, to count the number of times that the function has been called.

this = optimsimplex_new ( "pfeffer" , x0 )

Creates a new simplex so that it is computed from Pfeffer's method, i.e. a relative delta for non-zero values and an absolute delta for zero values.

The following is a complete list of available calling sequences.

this = optimsimplex_new ( "pfeffer" , x0 , fun )
this = optimsimplex_new ( "pfeffer" , x0 , fun , deltausual )
this = optimsimplex_new ( "pfeffer" , x0 , fun , deltausual , deltazero )
[ this , data ] = optimsimplex_new ( "pfeffer" , x0 , fun , deltausual , deltazero , data )
this

The current simplex object.

x0

the initial point, as a row vector.

fun

optional, the function to compute at vertices.

The function is expected to have the following input and output arguments :

function y=myfunction(x)
deltausual

optional, the absolute delta for non-zero values. The default value is 0.05.

deltazero

optional, the absolute delta for zero values. The default value is 0.0075.

data

optional, user-defined data passed to the function.

If data is provided, it is passed to the callback function both as an input and output argument. In that case, the function must have the following header :

function [y, data]=myfunction(x, data)

The data input parameter may be used if the function uses some additionnal parameters. It is returned as an output parameter because the function may modify the data while computing the function value. This feature may be used, for example, to count the number of times that the function has been called.

this = optimsimplex_new ( "randbounds" , x0 , fun , boundsmin , boundsmax , nbpoints )

Creates a new simplex so that it is computed by taking the bounds into account with random scaling. The number of vertices in the simplex is arbitrary.

The following is a complete list of available calling sequences.

[ this , data ] = optimsimplex_new ( "randbounds" , x0 , fun , boundsmin , boundsmax , nbpoints , data )
this

The current simplex object.

x0

the initial point, as a row vector. It is the first vertex in the simplex.

fun

optional, the function to compute at vertices.

The function is expected to have the following input and output arguments :

function y=myfunction(x)
boundsmin

array of minimum bounds

boundsmax

array of maximum bounds

Each component ix =1,n of the vertex #k = 2,nbve is computed from the formula :

x ( k , ix ) = boundsmin( ix ) + rand() * (boundsmax( ix ) - boundsmin( ix ))
nbpoints

total number of points in the simplex

data

optional, user-defined data passed to the function.

If data is provided, it is passed to the callback function both as an input and output argument. In that case, the function must have the following header :

function [y, data]=myfunction(x, data)

The data input parameter may be used if the function uses some additionnal parameters. It is returned as an output parameter because the function may modify the data while computing the function value. This feature may be used, for example, to count the number of times that the function has been called.

this = optimsimplex_new ( "spendley" , x0 )

Creates a new simplex so that it is computed from Spendley's et al. method, i.e. a regular simplex made of nbve = n+1 vertices.

The following is a complete list of available calling sequences.

this = optimsimplex_new ( "spendley" , x0 , fun )
this = optimsimplex_new ( "spendley" , x0 , fun , len )
[ this , data ] = optimsimplex_new ( "spendley" , x0 , fun , len, data )
this

The current simplex object.

x0

the initial point, as a row vector.

fun

optional, the function to compute at vertices.

The function is expected to have the following input and output arguments :

function y=myfunction(x)
len

optional, the length of the simplex. The default length is 1.0.

data

optional, user-defined data passed to the function.

If data is provided, it is passed to the callback function both as an input and output argument. In that case, the function must have the following header :

function [y, data]=myfunction(x, data)

The data input parameter may be used if the function uses some additionnal parameters. It is returned as an output parameter because the function may modify the data while computing the function value. This feature may be used, for example, to count the number of times that the function has been called.

this = optimsimplex_new ( "oriented" , simplex0 , fun )

Returns a new oriented simplex, in sorted order. The new simplex has the same sigma- length of the base simplex, but is "oriented" depending on the function value. The created simplex may be used, as Kelley suggests, for a restart of Nelder-Mead algorithm.

The following is a complete list of available calling sequences.

[ this , data ] = optimsimplex_new ( "oriented" , simplex0 , fun, data )
this

The current simplex object.

simplex0

the base simplex

fun

optional, the function to compute at vertices.

The function is expected to have the following input and output arguments :

function y=myfunction(x)
data

optional, user-defined data passed to the function.

If data is provided, it is passed to the callback function both as an input and output argument. In that case, the function must have the following header :

function [y, data]=myfunction(x, data)

The data input parameter may be used if the function uses some additionnal parameters. It is returned as an output parameter because the function may modify the data while computing the function value. This feature may be used, for example, to count the number of times that the function has been called.

this = optimsimplex_destroy (this)

Destroy the given object.

this

The current simplex object.

this = optimsimplex_setall ( this , simplex )

Set all the coordinates and and the function values of all the vertices.

this

The current simplex object.

simplex

the simplex to set.

The given matrix is expected to be a nbve x n+1 matrix where n is the dimension of the space, nbve is the number of vertices and with the following content (where the data is organized by row with function value first, and x coordinates)

  • simplex(k,1) is the function value of the vertex #k, with k = 1 , nbve

  • simplex(k,2:n+1) is the coordinates of the vertex #k, with k = 1 , nbve

this = optimsimplex_setallfv ( this , fv )

Set all the function values of all the vertices. The vertex #k is expected to be stored in fv(k) with k = 1 , nbve. The fv input argument is expected to be a row vector.

this

The current simplex object.

fv

the array of function values

this = optimsimplex_setallx ( this , x )

Set all the coordinates of all the vertices. The vertex #k is expected to be stored in x(k,1:n) with k = 1 , nbve

this

The current simplex object.

x

the coordinates of the vertices.

this = optimsimplex_setfv ( this , ive , fv )

Set the function value at given index and // returns an updated simplex.

this

The current simplex object.

ive

vertex index

fv

the function value

this = optimsimplex_setn ( this , n )

Set the dimension of the space of the simplex.

this

The current simplex object.

n

the dimension

this = optimsimplex_setnbve ( this , nbve )

Set the number of vertices of the simplex.

this

The current simplex object.

nbve

the number of vertices

this = optimsimplex_setve ( this , ive , fv , x )

Sets the coordinates of the vertex and the function value at given index in the current simplex.

this

The current simplex object.

ive

the vertex index

fv

the function value

x

the coordinates of the point, as a row vector.

this = optimsimplex_setx ( this , ive , x )

Set the coordinates of the vertex at given index, as a row vector, into the current simplex.

this

The current simplex object.

ive

the vertex index

x

the coordinates of the point, as a row vector

simplex = optimsimplex_getall ( this )

Returns all the coordinates of all the vertices and the function values in the same matrix.

this

The current simplex object.

simplex

the simplex data.

The simplex matrix has size nbve x n+1, and is organized by row by row as follows :

  • simplex(k,1) is the function value of the vertex #k, with k = 1 , nbve

  • simplex(k,2:n+1) is the coordinates of the vertex #k, with k = 1 , nbve

fv = optimsimplex_getallfv ( this )

Returns all the function values of all the vertices, as a row vector.

this

The current simplex object.

fv

The array of function values. The function value of vertex #k is stored in fv(k) with k = 1 , nbve.

x = optimsimplex_getallx ( this )

Returns all the coordinates of all the vertices.

this

The current simplex object.

x

the coordinates.

The vertex #k is stored in x(k,1:n) with k = 1 , nbve.

fv = optimsimplex_getfv ( this , ive )

Returns the function value at given index

this

The current simplex object.

ive

the vertex index

n = optimsimplex_getn ( this )

Returns the dimension of the space of the simplex

this

The current simplex object.

nbve = optimsimplex_getnbve ( this )

Returns the number of vertices in the simplex.

this

The current simplex object.

vertex = optimsimplex_getve ( this , ive )

Returns the vertex at given index as a tlist, with fields n, x and fv

this

The current simplex object.

ive

the vertex index

x = optimsimplex_getx ( this , ive )

Returns the coordinates of the vertex at given index, as a row vector.

this

The current simplex object.

ive

the vertex index

sicenter = optimsimplex_center ( this )

Returns the center of the given simplex

this

The current simplex object.

optimsimplex_check ( this )

Check the consistency of the internal data. Generates an error if necessary.

this

The current simplex object.

[ this , data ] = optimsimplex_computefv ( this , fun , data )

Set the values of the function at vertices points.

this

The current simplex object.

fun

optional, the function to compute at vertices.

The function is expected to have the following input and output arguments :

function y=myfunction(x)
data

optional, user-defined data passed to the function.

If data is provided, it is passed to the callback function both as an input and output argument. In that case, the function must have the following header :

function [y, data]=myfunction(x, data)

The data input parameter may be used if the function uses some additionnal parameters. It is returned as an output parameter because the function may modify the data while computing the function value. This feature may be used, for example, to count the number of times that the function has been called.

df = optimsimplex_deltafv ( this )

Returns the vector of difference of function values with respect to the function value at vertex #1.

this

The current simplex object.

dfm = optimsimplex_deltafvmax ( this )

Returns the difference of function value between the high and the low vertices. It is expected that the vertex #1 is associated with the smallest function value and that the vertex #nbve is associated with the highest function value. Since vertices are ordered, the high is greater than the low.

this

The current simplex object.

m = optimsimplex_dirmat ( this )

Returns the n x n matrix of simplex directions i.e. the matrix of differences of vertices coordinates with respect to the vertex #1.

this

The current simplex object.

m = optimsimplex_fvmean ( this )

Returns the mean of the function value on the simplex.

this

The current simplex object.

sd = optimsimplex_fvstdev ( this )

Returns the standard deviation of the function value on the simplex.

this

The current simplex object.

v = optimsimplex_fvvariance ( this )

Returns the variance of the function value on the simplex.

this

The current simplex object.

g = optimsimplex_gradientfv ( this , fun , method )

Returns the simplex gradient of the function.

The following is a complete list of available calling sequences.

g = optimsimplex_gradientfv ( this , fun , method )
[ g , data ] = optimsimplex_gradientfv ( this , fun , method , data )
this

The current simplex object.

fun

optional, the function to compute at vertices.

The function is expected to have the following input and output arguments :

function y=myfunction(x)
method

optional, the method to use to compute the simplex gradient. Two methods are available : "forward" or "centered". The forward method uses the current simplex to compute the simplex gradient. The centered method creates an intermediate reflected simplex and computes the average.

If not provided, the default method is "forward".

data

optional, user-defined data passed to the function.

If data is provided, it is passed to the callback function both as an input and output argument. In that case, the function must have the following header :

function [y, data]=myfunction(x, data)

The data input parameter may be used if the function uses some additionnal parameters. It is returned as an output parameter because the function may modify the data while computing the function value. This feature may be used, for example, to count the number of times that the function has been called.

[ r , data ] = optimsimplex_reflect ( this , fun , data )

Returns a new simplex by reflexion of current simplex, by reflection with respect to the first vertex in the simplex. This move is used in the centered simplex gradient.

this

The current simplex object.

fun

optional, the function to compute at vertices.

The function is expected to have the following input and output arguments :

function y=myfunction(x)
data

optional, user-defined data passed to the function.

If data is provided, it is passed to the callback function both as an input and output argument. In that case, the function must have the following header :

function [y, data]=myfunction(x, data)

The data input parameter may be used if the function uses some additionnal parameters. It is returned as an output parameter because the function may modify the data while computing the function value. This feature may be used, for example, to count the number of times that the function has been called.

[ this , data ] = optimsimplex_shrink ( this , fun , sigma , data )

Shrink the simplex with given coefficient sigma and returns an updated simplex. The shrink is performed with respect to the first point in the simplex.

this

The current simplex object.

fun

optional, the function to compute at vertices.

The function is expected to have the following input and output arguments :

function y=myfunction(x)
sigma

optional, the shrinkage coefficient. The default value is 0.5.

data

optional, user-defined data passed to the function.

If data is provided, it is passed to the callback function both as an input and output argument. In that case, the function must have the following header :

function [y, data]=myfunction(x, data)

The data input parameter may be used if the function uses some additionnal parameters. It is returned as an output parameter because the function may modify the data while computing the function value. This feature may be used, for example, to count the number of times that the function has been called.

ssize = optimsimplex_size ( this , method )

Returns the size of the simplex.

this

The current simplex object.

method

optional, the method to use to compute the size.

The available methods are the following :

  • "sigmaplus" (this is the default)

    The sigmamplus size is the maximum 2-norm length of the vector from each vertex to the first vertex. It requires one loop over the vertices.

  • "sigmaminus"

    The sigmaminus size is the minimum 2-norm length of the vector from each vertex to the first vertex. It requires one loop over the vertices.

  • "Nash"

    The "Nash" size is the sum of the norm of the norm-1 length of the vector from the given vertex to the first vertex. It requires one loop over the vertices.

  • "diameter"

    The diameter is the maximum norm-2 length of all the edges of the simplex. It requires 2 nested loops over the vertices.

this = optimsimplex_sort ( this )

Sorts the simplex with increasing function value order so that the smallest function value is at vertex #1

this

The current simplex object.

str = optimsimplex_tostring ( this )

Returns the current simplex as a string.

this

The current simplex object.

cen = optimsimplex_xbar ( this , iexcl )

Returns the center of n vertices, by excluding the vertex with index iexcl. Returns a row vector.

this

The current simplex object.

iexcl

the index of the vertex to exclude in center computation. The default value of iexcl is the number of vertices : in that case, if the simplex is sorted in increasing function value order, the worst vertex is excluded.

Example : Creating a simplex with given vertices coordinates

In the following example, one creates a simplex with known vertices coordinates. The function values at the vertices are unset.

coords = [
    0.    0.  
    1.    0.  
    0.    1.  
];
s1 = optimsimplex_new ( coords );
optimsimplex_getallx ( s1 )
optimsimplex_getn(s1)
optimsimplex_getnbve (s1)
s1 = optimsimplex_destroy(s1);

Example : Creating a simplex with randomized bounds

In the following example, one creates a simplex with in the 2D domain [-5 5]^2, with [-1.2 1.0] as the first vertex. One uses the randomized bounds method to generate a simplex with 5 vertices. The function takes an additionnal argument mystuff, which is counts the number of times the function is called. After the creation of the simplex, the value of mystuff.nb is 5, which is the expected result because there is one function call by vertex.

function y=rosenbrock(x)
  y = 100*(x(2)-x(1)^2)^2+(1-x(1))^2;
endfunction
function [y, mystuff]=mycostf(x, mystuff)
  y = rosenbrock(x);
  mystuff.nb = mystuff.nb + 1
endfunction

mystuff = tlist(["T_MYSTUFF","nb"]);
mystuff.nb = 0;
[ s1 , mystuff ] = optimsimplex_new ( "randbounds" , [-1.2 1.0], mycostf, ...
  [-5.0 -5.0] , [5.0 5.0], 5 , mystuff );
s1
mprintf("Function evaluations: %d\n",mystuff.nb)
s1 = optimsimplex_destroy ( s1 );

Initial simplex strategies

In this section, we analyse the various initial simplex which are provided in this component.

It is known that direct search methods based on simplex designs are very sensitive to the initial simplex. This is why the current component provides various ways to create such an initial simplex.

The first historical simplex-based algorithm is the one presented in "Sequential Application of Simplex Designs in Optimisation and Evolutionary Operation" by W. Spendley, G. R. Hext and F. R. Himsworth. The "spendley" simplex creates the regular simplex which is presented in the paper.

The "randbounds" simplex is due to M.J. Box in "A New Method of Constrained Optimization and a Comparison With Other Methods".

Pfeffer's method is an heuristic which is presented in "Global Optimization Of Lennard-Jones Atomic Clusters" by Ellen Fan. It is due to L. Pfeffer at Stanford and it is used in fminsearch.

TODO

  • implement reflection and expansion as in multidimensional search by Torczon

  • turn optimsimplex_reflect into a proper constructor, i.e. an option of the the optimsimplex_new function. Another possibility is to reflect "in place" as in the optimsimplex_shrink function (but in this case we must provide a "copy" constructor from current simplex before reflecting it).

Authors

Michael Baudin - INRIA - 2008-2009

Michael Baudin - Digiteo - 2009-2011

Bibliography

“Sequential Application of Simplex Designs in Optimisation and Evolutionary Operation”, Spendley, W. and Hext, G. R. and Himsworth, F. R., American Statistical Association and American Society for Quality, 1962

"A Simplex Method for Function Minimization", Nelder, J. A. and Mead, R. The Computer Journal, January, 1965, 308--313

"A New Method of Constrained Optimization and a Comparison With Other Methods", M. J. Box, The Computer Journal 1965 8(1):42-52, 1965 by British Computer Society

"Iterative Methods for Optimization", C.T. Kelley, 1999, Chapter 6., section 6.2

"Compact Numerical Methods For Computers - Linear Algebra and Function Minimization", J.C. Nash, 1990, Chapter 14. Direct Search Methods

"Sequential Application of Simplex Designs in Optimisation and Evolutionary Operation", W. Spendley, G. R. Hext, F. R. Himsworth, Technometrics, Vol. 4, No. 4 (Nov., 1962), pp. 441-461, Section 3.1

"A New Method of Constrained Optimization and a Comparison With Other Methods", M. J. Box, The Computer Journal 1965 8(1):42-52, 1965 by British Computer Society

“Detection and Remediation of Stagnation in the Nelder--Mead Algorithm Using a Sufficient Decrease Condition”, SIAM J. on Optimization, Kelley,, C. T., 1999

" Multi-Directional Search: A Direct Search Algorithm for Parallel Machines", by E. Boyd, Kenneth W. Kennedy, Richard A. Tapia, Virginia Joanne Torczon,, Virginia Joanne Torczon, 1989, Phd Thesis, Rice University

"Grid Restrained Nelder-Mead Algorithm", Árpád Bũrmen, Janez Puhan, Tadej Tuma, Computational Optimization and Applications, Volume 34 , Issue 3 (July 2006), Pages: 359 - 375

"A convergent variant of the Nelder-Mead algorithm", C. J. Price, I. D. Coope, D. Byatt, Journal of Optimization Theory and Applications, Volume 113 , Issue 1 (April 2002), Pages: 5 - 19,

"Global Optimization Of Lennard-Jones Atomic Clusters", Ellen Fan, Thesis, February 26, 2002, McMaster University

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Copyright (c) 1989-2007 (ENPC)
with contributors
Last updated:
Wed Oct 05 12:10:58 CEST 2011