optim - non-linear optimization routine

Calling Sequence

[f,xopt]=optim(costf,x0)

[f,[xopt,[gradopt,[work]]]]=optim(costf,[contr],x0,['algo'],[df0,[mem]],

[work],[stop],['in'],[imp=iflag])

Parameters

• costf : external, i.e Scilab function list or string (costf is the cost function: see below its calling sequence (Scilab or Fortran)). See also external for details about external functions.
• x0 : real vector (initial value of variable to be minimized).
• f : value of optimal cost (f=costf(xopt))
• xopt : best value of x found.
• contr : 'b',binf,bsup with binf and bsup real vectors with same dimension as x0. binf and bsup are lower and upper bounds on x.
• "algo" : 'qn' or 'gc' or 'nd' . This string stands for quasi-Newton (default), conjugate gradient or non-differentiable respectively. Note that 'nd' does not accept bounds on x ).
• df0 : real scalar. Guessed decreasing of f at first iteration. (df0=1 is the default value).
• mem : integer, number of variables used to approximate the Hessian, (algo='gc' or 'nd'). Default value is around 6.
• stop : sequence of optional parameters controlling the convergence of the algorithm. stop= 'ar',nap, [iter [,epsg [,epsf [,epsx]]]]
• "ar" : reserved keyword for stopping rule selection defined as follows:
• nap : maximum number of calls to costf allowed.
• iter : maximum number of iterations allowed.
• epsg : threshold on gradient norm.
• epsf : threshold controlling decreasing of f
• epsx : threshold controlling variation of x. This vector (possibly matrix) of same size as x0 can be used to scale x.
• "in" : reserved keyword for initialization of parameters used when costf in given as a Fortran routine (see below).
• "imp=iflag" : named argument used to set the trace mode. iflag=0 nothing (execpt errors) is reported, iflag=1 initial and final reports, iflag=2 adds a report per iteration, iflag>2 add reports on linear search. Warning, most of these reports are written on the Scilab standard output.
• gradopt : gradient of costf at xopt
• work : working array for hot restart for quasi-Newton method. This array is automatically initialized by optim when optim is invoked. It can be used as input parameter to speed-up the calculations.

Description

Non-linear optimization routine for programs without constraints or with bound constraints:

min costf(x) w.r.t x.
   

costf is an "external" i.e function, or list or Fortran routine (see "external"). This external must return f (costf(x)) and g (gradient of costf) given x.

If costf is a function, the calling sequence for costf must be:

[f,g,ind]=costf(x,ind).
   

Here, costf is a function which returns f, value (real number) of cost function at x, and g, gradient vector of cost function at x. The variable ind is used by optim and is described below.

If ind=2 (resp. 3, 4), costf must provide f (resp. g, f and g).

If ind=1 nothing is computed (used for display purposes only).

On output, ind<0 means that f cannot be evaluated at x and ind=0 interrupts the optimization.

If costf is a character string, it refers to the name of a Fortran routine which must be linked to Scilab (see examples in the routines foptim.f and e.g. genros.f in the directory SCIDIR/default)

Dynamic link of Fortran routine is also possible (help link).

Here, the generic calling sequence for the Fortran subroutine is: function costf(ind,n,x,f,g,ti,tr,td)

ind has the same meaning as above if set to 0,1,2 but the values ind=10 and ind=11 are now valid. These values are used for initializations (see below).

n is the dimension of x, x is an n vector, ti,tr,td are working arrays.

The Fortran function costf must return f and the vector g, given x, ind, n, ti, tr, td.

If costf is given as a Fortran routine, it is possible to initialize parameters or to send Scilab variables to this routine.

This facility is managed by the parameter 'in.

If the string 'in' is present, initialization is done by Fortran: optim makes two calls to the Fortran function costf, once with ind=10 and once with ind=11. In this case, for ind=10, costf must set the dimensions nti, ntr, ntd of ti, tr, td in the common/nird/nti, ntr, ntd and, for ind=11, costf must initialize the vectors ti , tr, td (integer vector, real vector, double precision vector respectively).

In the calling sequence of optim, the string 'in' can be replaced by 'ti', valti ,'td' , valtd. Then, the Fortran function costf(ind, x, f, g, ti, tr, td) is evaluated with ti=valti and td=valtd whatever the value of ind. Thus, the Scilab variables valti and valtd (integer vector and real vector) are sent to the Fortran function costf.

It is also possible to save the content of of the working arrays ti and td. This is possible by adding the strings 'si' and/or 'sd' at the ned of the calling sequence of optim. Then, the output variables must be: [f,[x,[g],[to]]],[ti],[td]].

Examples

xref=[1;2;3];x0=[1;-1;1]
deff('[f,g,ind]=cost(x,ind)','f=0.5*norm(x-xref)^2,g=x-xref');
[f,xopt]=optim(cost,x0)      //Simplest call
[f,xopt,gopt]=optim(cost,x0,'gc')  // By conjugate gradient
[f,xopt,gopt]=optim(cost,x0,'nd')  //Seen as non differentiable
[f,xopt,gopt]=optim(cost,'b',[-1;0;2],[0.5;1;4],x0) //  Bounds on x
[f,xopt,gopt]=optim(cost,'b',[-1;0;2],[0.5;1;4],x0,'gc') //  Bounds on x
[f,xopt,gopt]=optim(cost,'b',[-1;0;2],[0.5;1;4],x0,'gc','ar',3)
// Here, 3 calls to cost are allowed.
// Now calling the Fortran subroutine "genros" in SCIDIR/default/Ex-optim.f
// See also the link function for dynamically linking an objective function
[f,xopt,gopt]=optim('genros',[1;2;3])    //Rosenbrock's function
   

See Also

external  quapro  linpro  datafit  leastsq  numdiff  derivative  NDcost