ndbase.simplex

simplex optimisation

Syntax

[pOpt,fVal,stat] = ndbase.simplex([],func,p0,Name,Value)

[pOpt,fVal,stat] = ndbase.simplex(dat,func,p0,Name,Value)

Description

[pOpt,fVal,stat] = ndbase.simplex([],func,p0,Name,Value) finds a minimum of a function of several parameters using the constrained simplex optimization algorithm by calling the unconstrained Matlab built-in algorithm fminsearch.

The function has two different modes, depending on the first input argument. If dat is empty, simplex minimizes the cost function func, that has the following header:

R2 = func(p)

If dat is a struct, simplex optimizes the model defined by func via least squares to the data stored in dat. In this case func has the following header:

yModel = func(x,p)

And the least square deviation is defined by:

$R^2 = \sum \frac{(y_{dat}-y_{fit})^2}{\sigma_{dat}^2}$

Note: If options is supplied, then TolX will apply to the transformed variables. All other fminsearch parameters should be unaffected. Variables which are constrained by both a lower and an upper bound will use a $\sin(x)$ transformation. Those constrained by only a lower or an upper bound will use a quadratic transformation, and unconstrained variables will be left alone. Variables may be fixed by setting their respective bounds equal. In this case, the problem will be reduced in size for fminsearch. The bounds are inclusive inequalities, which admit the boundary values themselves, but will not permit any function evaluations outside the bounds. These constraints are strictly followed. If your problem has an exclusive (strict) constraint which will not admit evaluation at the bound itself, then you must provide a slightly offset bound. An example of this is a function which contains the log of one of its parameters. If you constrain the variable to have a lower bound of zero, then simplex may try to evaluate the function exactly at zero.

Examples

Example usage on the rosen function.

rosen = @(x) (1-x(1)).^2 + 105*(x(2)-x(1).^2).^2

Unconstrained optimisation:

fminsearch(rosen,[3 3])

Output

    1.0000    1.0000

Constrained optimisation:

ndbase.simplex([],rosen,[3 3],'lb',[2 2],'ub',[])

Output

    2.0000    4.0000

Input Arguments

dat

Either empty or contains data to be fitted stored in a structure with fields:

dat.x vector of $N$ independent variables,
dat.y vector of $N$ data values to be fitted,
dat.e vector of $N$ standard deviation (positive numbers) used to weight the fit. If zero or missing 1/dat.y^2 will be assigned to each point.

func

Function handle with one of the following definition:

R2 = func(p) if dat is empty,
y = func(x,p) if dat is a struct. Here x is a vector of $N$ independent variables, p are the $M$ parameters to be optimized and y is the simulated model, R2 is the value to minimize.

Name-Value Pair Arguments

'lb': Vector with $N$ elements, lower boundary of the parameters. Default value is -inf.
'ub': Vector with $N$ elements, upper boundary of the parameters. Default value is inf.
'MaxIter': Maximum number of iterations, default value is $100M$ .
'MaxFunEvals': Maximum number of function evaluations, default value is $1000M$ . NOT IMPLEMENTED!
'TolX': Convergence tolerance for parameters, default value is $10^{-3}$ .
'Display': Level of information to print onto the Command Line during optimisation. Possible values are 'off', 'notify' and 'iter'. Default value is 'off'.
'TolFun': Convergence tolerance on the R2 value (return value of func or the weighted least square deviation from data). Default value is $10^{-3}$ .