====== Minimization of functions ======

The minimization of general functions of one or more variables can be carried out using the command:

''MINIMIZE'', //func, x$_1$//[, //x$_2$, x$_3$, …//]

where //func// represents a function of up to 50 variables //x$_1$, x$_2$, …//. Two different optimization methods can be selected as described below which do or do not use numerical derivative information.

The optimization method, as well as finer control over //func//, can be chosen using the ''METHOD'' directive

''METHOD'', //key// [, //key1=value, key2=value, …//] [minimize:method]

where //key// defines the optimization method. Valid options for //key// are:

  * **''BFGS''** Broyden-Fletcher-Goldfarb-Shanno conjugate gradient method, which uses numerical gradients (default)
  * **''SIMPLEX''** Downhill simplex method, which uses only function evaluations

Options to these methods, //key1, key2, …//, are:

  * **''VARSCALE''=//vscale//** Optimization in space of scaled variables.\\
//vscale=0// no scaling (not recommended)\\
//vscale=1// optimization in the space of ln($x$)\\
//vscale=2// optimization in space of initial value scaling, e.g., $x_1/x_{1i}$ (default)
  * **''THRESH''=//thresh//** Required accuracy of either the gradient (BFGS) or parameters (SIMPLEX). The default is $1 \cdot 10^{-4}$ for BFGS and $1 \cdot 10^{-2}$ for SIMPLEX. Note that previously this pertained to the function value in the Simplex case.
  * **''VSTEP''=//epsd//** Step size for numerical gradients (BFGS) or initial SIMPLEX vertices
  * **''PROC''=//procname//** Specifies the procedure to be executed in each optimization step. This defines a complete function evaluation (if needed, numerical gradients will be evaluated using this procedure as well)
  * **''STARTCMD''=//command//** Specifies a start command. In each optimization step all input beginning with //command// to the current ''MINIMIZE'' is processed.

Miscellaneous directives (separated by semicolons or linebreaks)

  * **''MAXIT'',//maxit//** maximum number of optimization cycles. The default is 30 for BFGS and 100 for SIMPLEX.

===== Examples =====

==== Geometry optimization ====

<code - examples/min_optgeo.inp>
***, Simple geometry optimization

basis=vdz

geometry={
O
H 1 r
H 1 r 2 theta}

r=1.8
theta=104

hf
mp2

{minimize,energy,r,theta}
---
</code>

==== Basis function optimizations ====

<code - examples/basisopt_simple.inp>
***, Optimization of 2 d functions

geometry={Ne}

dexp=[2.0,1.0]

basis={
sp,Ne,vdz;c;
d,Ne,dexp(1),dexp(2)
}

hf
mp2
eval=energy

minimize,eval,dexp(1),dexp(2)
---
</code>

<code - examples/basisopt_proc.inp>
***, Optimization of 2 d functions

geometry={Ne}

dexp=[2.0,1.0]

{minimize,eval,dexp(1),dexp(2)
method,bfgs,varscale=1,thresh=1e-5,proc=optd}


proc optd

basis={
sp,Ne,vdz;c;
d,Ne,dexp(1),dexp(2)
}

hf
mp2
eval=energy

endproc
</code>

<code - examples/basisopt_cv.inp>
***, MP2 optimization of core-valence cc-pCVDZ functions

geometry={Ne}

sexp=20.
pexp=30.

{minimize,ecv,sexp,pexp
method,bfgs,varscale=1,thresh=1e-5,proc=myopt}


proc myopt

basis={
spd,Ne,vdz;c;
s,Ne,sexp
p,Ne,pexp
}

hf
{mp2;core,1}
eval=energy

{mp2;core,0}
eall=energy

ecv=eall-eval

endproc
</code>

==== Intersection points ====

<code - examples/min_intersect.inp>
***, Try to find intersection seam of singlet-triplet CH2 as function of angle

basis=vdz

geometry={
C
H 1 r
H 1 r 2 theta}

r=1.6
angles=[70,75,80,85,90]

do i=1,#angles

 theta=angles(i)

 {minimize,ediff,r
 method,bfgs,thresh=1e-5,proc=findit}

 ropt(i)=r
 eseam(i)=etriplet
 converge(i)=ediff

enddo

table,angles,ropt,eseam,converge
digit,0,3,5,6


proc findit

 {hf;wf,8,2,2}
 etriplet=energy

 {hf;wf,8,1,0}
 esinglet=energy

 ediff=(etriplet-esinglet)**2

endproc
---
</code>