Mathematica NMinimize Program Running Slow - What's Wrong?

[shafieza_garl]

i try run my program for NMinimize.it took a long time to solve.What is the problem of my coding?

 

[CompuChip]

It would be nicer if you just copy/pasted the code, using

 and [/ code] tags like
[code]
this

Then I can plug it into my own Mathematica and its easier to analyse.

 

[shafieza_garl]

ok..this is the code...

 a = 1/4; b = 4044 + 4.0/9; c = 66 + 2.0/3; d = 
 15.0/1000; x0 = 0.0; x3 = 5.0; n = 3; K = 9.0; c2 = 0.1; \[Alpha] = \
0.25; p2 = 35.0; c1 = 25.0; i2 = 0.04; i1 = 0.05; \[Theta] = 0.03;
R[x_, y_] = 
  Integrate[
   a*(E^(-d*t)*(E^(d*t)*(b - c*t) + E^(d*y)*(-b + c*y)))^2, {t, x, 
    y}]; 
F[x_, y_] = 
  Integrate[
   a*(E^(d*t)*(E^(d*t)*(b - c*t) + E^(d*y)*(-b + c*y)))^2, {t, 
    y*(1 - \[Alpha]) + x*\[Alpha], y}];
H[x_, y_] = 
  0.5*Integrate[
    Integrate[
     E^(d*t)*(6 + 
        t)*((E^(d*t)*(b - c*t) + E^(d*y)*(-b + c*y))^2)^0.5, {t, x, 
      t}], {t, x, y + x*\[Alpha] - y*\[Alpha]}];
NMinimize[{n*K + c2*(R[x0, x1] + R[x1, x2] + R[x2, x3]) + 
   c1*i1*(F[x0, x1] + F[x1, x2] + F[x2, x3]) - 
   p2*i2*(H[x0, x1] + H[x1, x2] + H[x2, x3]), 
  x0 <= x1 && x1 <= x2 && x2 <= x3}, {x0, x1, x2, x3}]

 

[CompuChip]

First of all, you should replace the assignment by a delayed assignment (replace = by := in R, F and H), otherwise x and y don't get substituted.

The main problem, however, is in your definition of H, I think.
Personally, I am not fond of an expression like

Integrate[f[t], {t, x, t}]

Since you seem only interested in numerical solutions, I rewrote your code to the following:

R[x_?NumericQ, y_?NumericQ] := 
 NIntegrate[
  a*(E^(-d*t)*(E^(d*t)*(b - c*t) + E^(d*y)*(-b + c*y)))^2, {t, x, y} //
    Evaluate]
F[x_?NumericQ, y_?NumericQ] := 
 NIntegrate[
  a*(E^(d*t)*(E^(d*t)*(b - c*t) + E^(d*y)*(-b + c*y)))^2, {t, 
    y*(1 - \[Alpha]) + x*\[Alpha], y} // Evaluate]
H[x_?NumericQ, y_?NumericQ] := 
  0.5*NIntegrate[
    E^(d*t)*(6 + 
       t)*((E^(d*t)*(b - c*t) + E^(d*y)*(-b + c*y))^2)^0.5, {\[Tau], 
     x, y + x*\[Alpha] - y*\[Alpha]}, {t, x, \[Tau]}];
FindMinimum[{n*K + c2*(R[x0, x1] + R[x1, x2] + R[x2, x3]) + 
   c1*i1*(F[x0, x1] + F[x1, x2] + F[x2, x3]) - 
   p2*i2*(H[x0, x1] + H[x1, x2] + H[x2, x3]), 
  x0 <= x1 <= x2 <= x3}, {{x1, 1}, {x2, 3}}]

It is not really working now, but that seems to be due some problems with the numerical integration... I don't have time to solve those now, but maybe this will help you speed things up

 

[Bill Simpson]

The H[x0, x1] + H[x1, x2] + H[x2, x3] is what is taking the time. Without that the minimization completes in a fraction of a second.

Changing the t to tau doesn't seem to help.

And he has things like ((expr)^2)^0.5, sort of like what they teach in FORTRAN when they are trying to simulate absolute value, but changing the 0.5 to 1/2 or using Sqrt or Abs, none of these make any significant change in the time. I've tried to understand what he is doing well enough to turn this into a double integral, rather than a nested pair of integrals, but I cannot be certain I've done that correctly, especially with his reuse of names, and this still doesn't seem to make any difference.

Getting rid of the decimal points he keeps putting back in each time he posts this question does speed things up by perhaps 1/3, but that is not addressing the overwhelming time sink that his H[x,y] calculation is.

H[x,y] isn't oscillating, discontinuous or going to infinity and appears to almost be a plane over the region he is interested in.