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# iFit/fminsimpsa

## PURPOSE

[MINIMUM,FVAL,EXITFLAG,OUTPUT] = fminsimpsa(FUN,PARS,[OPTIONS],[CONSTRAINTS]) simplex/simulated annealing

## SYNOPSIS

function [pars,fval,exitflag,output] = fminsimpsa(varargin)

## DESCRIPTION

``` [MINIMUM,FVAL,EXITFLAG,OUTPUT] = fminsimpsa(FUN,PARS,[OPTIONS],[CONSTRAINTS]) simplex/simulated annealing

fminsimpsa finds a minimum of a function of several variables using an algorithm
that is based on the combination of the non-linear simplex and the simulated
annealing algorithm (the SIMPSA algorithm, Cardoso et al., 1996).
In this paper, the algorithm is shown to be adequate for the global optimi-
zation of an example set of unconstrained and constrained NLP functions.

Calling:
fminsimpsa(fun, pars) asks to minimize the 'fun' objective function with starting
parameters 'pars' (vector)
fminsimpsa(fun, pars, options) same as above, with customized options (optimset)
fminsimpsa(fun, pars, options, fixed)
is used to fix some of the parameters. The 'fixed' vector is then 0 for
free parameters, and 1 otherwise.
fminsimpsa(fun, pars, options, lb, ub)
is used to set the minimal and maximal parameter bounds, as vectors.
fminsimpsa(fun, pars, options, constraints)
where constraints is a structure (see below).
fminsimpsa(problem) where problem is a structure with fields
problem.objective:   function to minimize
problem.x0:          starting parameter values
problem.options:     optimizer options (see below)
problem.constraints: optimization constraints
fminsimpsa(..., args, ...)
sends additional arguments to the objective function
criteria = FUN(pars, args, ...)

Example:
banana = @(x)100*(x(2)-x(1)^2)^2+(1-x(1))^2;
[x,fval] = fminsimpsa(banana,[-1.2, 1])

Input:
FUN is the function to minimize (handle or string): criteria = FUN(PARS)
It needs to return a single value or vector.

PARS is a vector with initial guess parameters. You must input an
initial guess. PARS can also be given as a single-level structure.

OPTIONS is a structure with settings for the optimizer,
compliant with optimset. Default options may be obtained with
o=fminsimpsa('defaults')
options.MinFunEvals sets the minimum number of function evaluations to reach
An empty OPTIONS sets the default configuration.

CONSTRAINTS may be specified as a structure
constraints.min= vector of minimal values for parameters
constraints.max= vector of maximal values for parameters
constraints.fixed= vector having 0 where parameters are free, 1 otherwise
constraints.step=  vector of maximal parameter changes per iteration
constraints.eval=  expression making use of 'p', 'constraints', and 'options'
and returning modified 'p'
or function handle p=@constraints.eval(p)
An empty CONSTRAINTS sets no constraints.

Additional arguments are sent to the objective function.

Output:
MINIMUM is the solution which generated the smallest encountered
value when input into FUN.
FVAL is the value of the FUN function evaluated at MINIMUM.
EXITFLAG return state of the optimizer
OUTPUT additional information returned as a structure.

Reference: Section 10.4 and 10.9 in "Numerical Recipes in C", ISBN 0-521-43108-5
Cardoso, Salcedo, Feyo de Azevedo and Bardosa, Comp. Chem Engng, 21 (1997) 1349
Kirkpatrick, J. Stat. Phys. 34 (1984) 975.
Contrib:   2006 Brecht Donckels, BIOMATH, brecht.donckels@ugent.be
Systems Biology Toolbox for MATLAB, 2005 Henning Schmidt, FCC, henning@fcc.chalmers.se [SIMPSA]