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| PLS_Toolbox Documentation: lmoptimizebnd | < lmoptimize | localmax > |
lmoptimizebnd
Purpose
Levenberg-Marquardt bounded non-linear optimization.
Synopsis
[x,fval,exitflag,out] =
lmoptimizebnd(fun,x0,xlow,xup,options,params)
Description
Starting at (x0)
LMOPTIMIZE finds (x) that minimizes the
function defined by the function handle (fun) where (x) has 
parameters. Inputs (xlow) and (xup) can be used to provide
lower and upper bounds on the solution (x). The function (fun) must supply the Jacobian and Hessian i.e. they
are not estimated by LMOPTIMIZEBND
(an example description is provided in the Algorithm Section of the function LMOPTIMIZE).
INPUTS:
(jacobian)
is a x1
vector of Jacobian values, and
(hessian) is a x matrix of Hessian values.
x0 = x1 initial guess of the function
parameters.
xlow = x1 vector of corresponding lower
bounds on (x). See options.alow. If an element
of xlow == -inf, the
corresponding parameter in (x)
is unbounded on the low side.
xup = x1 vector of corresponding upper
bounds on (x). See options.aup. If an element
of xup == inf, the
corresponding parameter in (x)
is unbounded on the high side.
OPTIONAL INPUTS:
OUTPUTS:
x = x1 vector of parameter value(s) at
the function minimum.
Algorithm
The algorithm is essentially the same as that discussed in LMOPTIMIZE and this section discusses only the two main differences between LMOPTIMIZEBND and LMOPTIMIZE.
The first difference is the addition of penalty functions used
to enforce bounding. For example, the objective function used in LMOPTIMIZE is 
, but the
objective function used by LMOPTIMIZEBND
is 
.
The penalty functions for upper, 
, and lower bounds, 
, are similar,
so only the lower penalty function is described.
Define 
as the lower boundary, 
a small number
(e.g. 0.001) and 
a large number [e.g. 
], then for a
single parameter the lower penalty function is given as
.
This function can be considered an external point function
because it is defined outside the feasible region (outside the boundaries). It
is continuous at the boundary and also has continuous first and second
derivatives. This is in contrast to internal point functions such as a log
function that is not continuous at the boundary [e.g. 
is not continuous]. The
first and second derivatives of the penalty function are given by
and
.
The external point penalty function does not guarantee that a step won’t move outside the boundaries into the infeasible region. It does, however provide a means for getting back inside the feasible region.
A second modification is included in the LMOPTIMIZEBND algorithm to
avoid large steps outside the feasible region. If a step 
is such that any 
are outside the
feasible region, the step size for those parameters is reduced. The reduction
is 90% the distance of that parameter to the boundary. This typically changes
the direction of the step .
Options
alow: [ ], x1 vector of penalty
weights for lower bound, {default = ones(N,1)}. If an element is zero, the corresponding parameter
in (x) is not bounded
on the low side.
aup: [ ], x1 vector of penalty
weights for upper bound, {default = ones(N,1)}. If an element is zero, the corresponding parameter
in (x) is not bounded
on the high side.
Examples
options = lmoptimize('options');
options.x = 'on';
options.display = 'off';
options.alow = [0 0]; %x(1) and x(2) unbounded on low side
options.aup = [1 0]; %x(1) bounded on high side and x(2)
% unbounded on high side
[x,fval,exitflag,out] = lmoptimize(@banana,x0,[0 0], ...
[0.9 0],options);
plot(out.x(:,1),out.x(:,2),'-o','color', ...
[0.4 0.7 0.4],'markersize',2,'markerfacecolor', ...
[0 0.5 0],'markeredgecolor',[0 0.5 0])
See Also
function_handle, lmoptimize
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