Differences

This shows you the differences between two versions of the page.

--- minimization_of_functions [2020/06/11 18:17] – external edit 127.0.0.1
+++ minimization_of_functions [2024/07/12 08:37] (current) – external edit 127.0.0.1
@@ Line 1: / Line 1: @@
+====== 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>