| Usage: |
min = nmsimpen(fname,cname,x0{,fder,opt})
|
| Input: |
| fname | string, name of the function to minimize.
The function should have just one
parameter x, which is either a n x 1 vector,
or a n x k matrix (for any k >= 1) containing
x = (x1,x2,...,xk);
xi (n x 1 vector; i=1..k) represent points
at which the function should be evaluated.
As a result, the function should return
a scalar or 1 x k vector, respectively.
|
| cname | string, name of a function evaluating equality constraints or
a list containing at least one of cname.eq and
cname.neq, both strings containing names of functions
evaluating equality and inequality constraints, respectively.
The functions should have just one
parameter x, which is either a n x 1 vector,
or a n x k matrix (for any k >= 1) containing
x = (x1,x2,...,xk);
xi (n x 1 vector; i=1..k) represent points
at which the constraints should be evaluated.
As a result, the functions should return
c x 1 vector (for c constraints), resp. c x k matrix in case of input matrix.
|
| x0 | n x 1 vector, the initial estimate of the minimum
|
| fder | optional string or scalar, reserved for future usage as follows:
derivatives of fname; there are several possible
formats:
1. string, name of the function for computing the gradient
of func; its output is a n x 1 vector
2. empty string; in this case the gradient will be
computed automatically using the quantlet nmgraddiff
with a default step h;
this possibility is used now for any input values of fder
3. scalar h; the gradient will be computed
automatically using the quantlet nmgraddiff with
the given step h
|
| opt.xtol | optional scalar, convergence tolerance for x;
default is xtol = 1e-5
|
| opt.ftol | optional scalar, convergence tolerance of the function value;
default is ftol = 1e-5
|
| opt.maxiter | optional scalar, maximal number of iterations;
default is maxiter = 10
|
| opt.method | optional string, name of the routine for unconstrained minimization;
default is method = "nmcongrad"
|
| Output: |
| min.xmin | n x 1 vector, minimum of fname (isolated to a fractional precision of tol) |
| min.fmin | scalar, minimal function value f(xmin) |
| min.penalty | list with elements cvaleq and cvalneq,
scalars containing the maximal absolute
value of the equality and inequality
constraints evaluated at xmin |
| min.iter | scalar, number of iterations performed |
- Example:
library("nummath")
; function definition
proc(p)=ftion(x)
p = x[1,]^2 + x[1,].*x[2,] - 1
endp
; constraint definition
proc(c)=constr(x)
c = x[1,] + x[2,]^2 - 2
endp
conmin = nmsimpen("ftion","constr",#(1,1))
conmin
; minimization of ftion under equality constraint constr
; starting from the initial point x =(1,1)
- Result:
Contents of conmin.xmin
[1,] -1.0137
[2,] 1.736
Contents of conmin.fmin
[1,] -1.7322
Contents of conmin.penalty.cvaleq
[1,] 1.4574e-06
Contents of conmin.penalty.cvalneq
[1,] 0
Contents of conmin.iter
[1,] 5
- Example:
library("nummath")
; function definition: x1^2 + x2^4
proc(p)=ftion(x)
p = x[1,]^2 + x[2,]^4
endp
; equality constraints: x1^4 + x2^2 = 3
proc(cval) = ceq(x)
cval = x[1,]^4 + x[2,]^2 - 3
endp
; inequality constraint: x1 + x2 >= 2
proc(cval) = cneq(x)
cval = x[1,] + x[2,] - 2
endp
eq = "ceq"
neq = "cneq"
conmin = nmsimpen("ftion",list(eq,neq),#(0,0))
conmin
; minimization with both equality and inequality
; constraints, starting at x1 = x2 = 0
- Result:
Contents of conmin.xmin
[1,] 1.2494
[2,] 0.75062
Contents of conmin.fmin
[1,] 1.8784
Contents of conmin.penalty.cvaleq
[1,] -1.0463e-06
Contents of conmin.penalty.cvalneq
[1,] 8.6553e-07
Contents of conmin.iter
[1,] 6
- Example:
library("nummath")
; function definition
proc(p)=ftion(x)
p = x[1,]^2 + x[2,]^4
endp
; two inequality constraint:
; x1 + x2 >= 2 and x1 <= 1, i.e., -x1 + 1 >= 0
proc(cval) = cneq(x)
cval =(x[1,] + x[2,] - 2) |(- x[1,] + 1)
endp
neq = "cneq"
conmin = nmsimpen("ftion",list(neq),#(0,0))
conmin
- Result:
Contents of conmin.xmin
[1,] 1
[2,] 1
Contents of conmin.fmin
[1,] 2
Contents of conmin.penalty.cvaleq
[1,] 0
Contents of conmin.penalty.cvalneq
[1,] 5.0487e-08
Contents of conmin.iter
[1,] 7
Notice that conmin.penalty.cvalneq contains
the maximum of absolute values of inequality
constraints evaluated at xmin. You can check it
it by entering command cneq(conmin.xmin)
that produces the result
Contents of cval
[1,] -5.0487e-08
[2,] 2.5243e-08