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/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}
---

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)
---

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

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

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
---