Direct search often runs faster if you vectorize the
objective and nonlinear constraint functions. This means your functions
evaluate all the points in a poll or search pattern at once, with
one function call, without having to loop through the points one at
a time. Therefore, the option Vectorized
= 'on'
works
only when CompletePoll
or CompleteSearch
is
also set to 'on'
. However, when you set Vectorized
= 'on'
, patternsearch
checks
that the objective and any nonlinear constraint functions give outputs
of the correct shape for vectorized calculations, regardless of the
setting of the CompletePoll
or CompleteSearch
options.
If there are nonlinear constraints, the objective function and the nonlinear constraints all need to be vectorized in order for the algorithm to compute in a vectorized manner.
Note:
Write your vectorized objective function or nonlinear constraint
function to accept a matrix with an arbitrary number of points. |
A vectorized objective function accepts a matrix as input and
generates a vector of function values, where each function value corresponds
to one row or column of the input matrix. patternsearch
resolves
the ambiguity in whether the rows or columns of the matrix represent
the points of a pattern as follows. Suppose the input matrix has m
rows
and n
columns:
If the initial point x0
is a column
vector of size m
, the objective function takes
each column of the matrix as a point in the pattern and returns a
row vector of size n
.
If the initial point x0
is a row
vector of size n
, the objective function takes
each row of the matrix as a point in the pattern and returns a column
vector of size m
.
If the initial point x0
is a scalar, patternsearch
assumes
that x0
is a row vector. Therefore, the input matrix
has one column (n
= 1, the input matrix is a vector),
and each entry of the matrix represents one row for the objective
function to evaluate. The output of the objective function in this
case is a column vector of size m
.
Pictorially, the matrix and calculation are represented by the following figure.
Structure of Vectorized Functions
For example, suppose the objective function is
$$f(x)={x}_{1}^{4}+{x}_{2}^{4}-4{x}_{1}^{2}-2{x}_{2}^{2}+3{x}_{1}-{x}_{2}/2.$$
If the initial vector x0
is
a column vector, such as [0;0]
, a function for
vectorized evaluation is
function f = vectorizedc(x) f = x(1,:).^4+x(2,:).^4-4*x(1,:).^2-2*x(2,:).^2 ... +3*x(1,:)-.5*x(2,:);
x0
is a row vector, such as [0,0]
,
a function for vectorized evaluation isfunction f = vectorizedr(x) f = x(:,1).^4+x(:,2).^4-4*x(:,1).^2-2*x(:,2).^2 ... +3*x(:,1)-.5*x(:,2);
Tip
If you want to use the same objective (fitness) function for
both pattern search and genetic algorithm, write your function to
have the points represented by row vectors, and write |
To minimize vectorizedc
, enter the following
commands:
options=psoptimset('Vectorized','on','CompletePoll','on'); x0=[0;0]; [x fval]=patternsearch(@vectorizedc,x0,... [],[],[],[],[],[],[],options)
Optimization terminated: mesh size less than options.TolMesh. x = -1.5737 1.0575 fval = -10.0088
Only nonlinear constraints need to be vectorized; bounds and linear constraints are handled automatically. If there are nonlinear constraints, the objective function and the nonlinear constraints all need to be vectorized in order for the algorithm to compute in a vectorized manner.
The same considerations hold for constraint functions as for
objective functions: the initial point x0
determines
the type of points (row or column vectors) in the poll or search.
If the initial point is a row vector of size k,
the matrix x passed to the constraint function
has k columns. Similarly, if the initial point
is a column vector of size k, the matrix of poll
or search points has k rows. The figure Structure of Vectorized Functions may make
this clear. If the initial point is a scalar, patternsearch
assumes
that it is a row vector.
Your nonlinear constraint function returns two matrices, one
for inequality constraints, and one for equality constraints. Suppose
there are n_{c} nonlinear inequality
constraints and n_{ceq} nonlinear
equality constraints. For row vector x0
, the constraint
matrices have n_{c} and n_{ceq} columns
respectively, and the number of rows is the same as in the input matrix.
Similarly, for a column vector x0
, the constraint
matrices have n_{c} and n_{ceq} rows
respectively, and the number of columns is the same as in the input
matrix. In figure Structure of Vectorized Functions, "Results"
includes both n_{c} and n_{ceq}.
Suppose that the nonlinear constraints are
$$\begin{array}{c}\frac{{x}_{1}^{2}}{9}+\frac{{x}_{2}^{2}}{4}\le 1\text{(theinteriorofanellipse),}\\ {x}_{2}\ge \mathrm{cosh}\left({x}_{1}\right)-1.\end{array}$$
Write a function for these constraints for row-form x0
as
follows:
function [c ceq] = ellipsecosh(x) c(:,1)=x(:,1).^2/9+x(:,2).^2/4-1; c(:,2)=cosh(x(:,1))-x(:,2)-1; ceq=[];
Minimize vectorizedr
(defined in Vectorized Objective Function) subject
to the constraints ellipsecosh
:
x0=[0,0]; options=psoptimset('Vectorized','on','CompletePoll','on'); [x fval]=patternsearch(@vectorizedr,x0,... [],[],[],[],[],[],@ellipsecosh,options)
Optimization terminated: mesh size less than options.TolMesh and constraint violation is less than options.TolCon. x = -1.3516 1.0612 fval = -9.5394