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A comparison of the convergence rates of Hestenes’ conjugate Gram-Schmidt method without derivatives with other numerical optimization methods

Md Nurul Raihen
Md Nurul Raihen
   | 03 jun 2024
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Article Category: Original Study
Publicado en línea: 03 jun 2024
Páginas: -
Recibido: 26 sept 2023
Aceptado: 12 feb 2024
DOI: https://doi.org/10.2478/ijmce-2025-0010
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© 2025 Md Nurul Raihen, published by Sciendo
This work is licensed under the Creative Commons Attribution 4.0 International License.

Fig. 1

Level curves of Rosenbrock’s Banana Valley function.
Level curves of Rosenbrock’s Banana Valley function.

j.ijmce-2025-0010.tab.002

Step 1: restart: Digits:=50;
Step 2: f [1] := (x, y)− > 2 ∗ (x − 1) − 400 ∗ x(y − x ∗ x);
Step 3: f [2] := (x, y)− > 200 ∗ (y − x ∗ x);
Step 4: h := 0.001
Step 5: t[1] := 0; x1[1] := −1.2; x2[1] := 1.0;
Step 6: Perform the following iteration:
for i from 1 to 100 do
  for j from 1 to 2 do
k1[ j] := f [ j](x1[i], x2[i])
  end for:
  for j from 1 to 2 do
k2[ j] := f [ j](x1[i] + (h/2) ∗ k1[1], x2[i] + (h/2) ∗ k1[2])
  end for:
  for j from 1 to 2 do
k3[ j] := f [ j](x1[i] + (h/2) ∗ k2[1], x2[i] + (h/2) ∗ k2[2])
  end for:
  for j from 1 to 2 do
k4[ j] := f [ j](x1[i] + h ∗ k3[1], x2[i] + h ∗ k3[2])
  end for: x1[i + 1] := x1[i] + (h/6) ∗ (k1[1] + 2 ∗ k2[1] + 2 ∗ k3[1] + k4[1]):
x2[i + 1] := x2[i] + (h/6) ∗ (k1[2] + 2 ∗ k2[2] + 2 ∗ k3[2] + k4[2]):
t[i + 1] := t[i] + h
end for:
Step 7: print:
for i from 1 by 10 to 100 do print(t[i], x1[i], x2[i]);
end for;

j.ijmce-2025-0010.tab.003

Step 1: restart: Digits:=150;
Step 2: with(Student[LinearAlgebra]):
Step 3: Perform the following iteration:
x :=< −1.2, 1 >:
for i from 1 to 9 do
Jacobian :=< −400 ∗ (x[2] − x[1]2) + 800 ∗ x[1]2 + 2, −400 ∗ x[1], 200 >:
Jacobian(−1):
x := x − (Jacobian(−1)). < −400 ∗ x[1] ∗ x[2] + 400 ∗ x[1]3 + 2 ∗ (x[1] − 1), 200 ∗ (x[2] − x[1]2) >:
print(x[1], x[2]):
y[i, 1] := x[1];y[i, 2] = x[2];
end for

j.ijmce-2025-0010.tab.004

Step 1: restart: Digits:=50;
Step 2: gradient[1] := (x, y)− > 2 ∗ (x − 1) − 400 ∗ x(y − x ∗ x);
Step 3: gradient[2] := (x, y)− > 200 ∗ (y − x ∗ x);
Step 4: NORMofGRADIENT:=sqrt((2 ∗ (x − 1) − 400 ∗ x∗ (y − x ∗ x))2 + (200 ∗ (y − x ∗ x))2);
Step 5: NORMALIZED[1]:=(x,y)−>(2*(1−x)+400*x*(y−x*x))sqrt((2*(x−1)−400*x*(y−x*x))2+(200*(y−x*x))2) {\rm{NORMALIZED}}[1]: = (x,y) - > \frac{{(2*(1 - x) + 400*x*(y - x*x))}}{{{\rm{sqrt}}({{(2*(x - 1) - 400*x*(y - x*x))}^2} + {{(200*(y - x*x))}^2})}} ;
Step 6: NORMALIZED[2]:=(x,y)−>200*(x*x−y)sqrt((2*(x−1)−400*x*(y−x*x))2+(200*(y−x*x))2) {\rm{NORMALIZED}}[2]: = (x,y) - > \frac{{200*(x*x - y)}}{{{\rm{sqrt}}({{(2*(x - 1) - 400*x*(y - x*x))}^2} + {{(200*(y - x*x))}^2})}} ;
Step 7: f[1]:=(x,y)−>(2*(1−x)+400*x*(y−x*x))sqrt((2*(x−1)−400*x*(y−x*x))2+(200*(y−x*x))2) f[1]: = (x,y) - > \frac{{(2*(1 - x) + 400*x*(y - x*x))}}{{{\rm{sqrt}}({{(2*(x - 1) - 400*x*(y - x*x))}^2} + {{(200*(y - x*x))}^2})}}
Step 8: f[2]:=(x,y)−>200*(x*x−y)sqrt((2*(x−1)−400*x*(y−x*x))2+(200*(y−x*x))2) f[2]: = (x,y) - > \frac{{200*(x*x - y)}}{{{\rm{sqrt}}({{(2*(x - 1) - 400*x*(y - x*x))}^2} + {{(200*(y - x*x))}^2})}}
Step 9: h := 0.005
Step 10: t[1] := 0; x1[1] := −1.2; x2[1] := 1.0;
Step 11: Perform the following iteration:
for i from 1 to 100 do
  for j from 1 to 2 do
k1[ j] := f [ j](x1[i], x2[i])
  end for:
  for j from 1 to 2 do
k2[ j] := f [ j](x1[i] + (h/2) ∗ k1[1], x2[i] + (h/2) ∗ k1[2])
  end for:
  for j from 1 to 2 do
k3[ j] := f [ j](x1[i] + (h/2) ∗ k2[1], x2[i] + (h/2) ∗ k2[2])
  end for:
  for j from 1 to 2 do
k4[ j] := f [ j](x1[i] + h ∗ k3[1], x2[i] + h ∗ k3[2])
  end for: x1[i + 1] := x1[i] + (h/6) ∗ (k1[1] + 2 ∗ k2[1] + 2 ∗ k3[1] + k4[1]):
x2[i + 1] := x2[i] + (h/6) ∗ (k1[2] + 2 ∗ k2[2] + 2 ∗ k3[2] + k4[2]):
t[i + 1] := t[i] + h
end for:
Step 12: print:
for i from 1 by 10 to 100 do print(t[i], x1[i], x2[i]);
end for;

j.ijmce-2025-0010.tab.005

Step 1: restart: Digits:=400;
Step 2: sigma := 1e−120; rho := 2 ∗ sigma; epsilon := .1e−60;
Step 3: u[1, 1] := 1; u[1, 2] := 0; u[2, 1] := 0; u[2, 2] := 1;
Step 4: p[1, 1] := u[1, 1]; p[1, 2] := u[1, 2];
Step 5: f := (x1, x2) → 100 ∗ (x2 − x1)2 + (x1 − 1)2;
Step 6: x[1, 1] := −1.2; x[1, 2] := 1;
Step 7: Perform the following iteration:
for j from 1 to 10 do
c[1] := (f (x[1, 1]) − sigma ∗ p[1, 1], x[1, 2] − sigma ∗ p[1, 2]) − f (x[1, 1] + sigma ∗ p[1, 1], x[1, 2] + sigma ∗
p[1, 2]))/(2 ∗ sigma):
d[1] := (f (x[1, 1] − sigma ∗ p[1, 1], x[1, 2] − sigma ∗ p[1, 2]) − 2 ∗ f (x[1, 1], x[1, 2]) + f (x[1, 1] + sigma ∗
p[1, 1], x[1, 2] + sigma ∗ p[1, 2]))/(sigma)2
a[1] := c[1]/d[1]:
x[2, 1] := x[1, 1] + a[1] ∗ p[1, 1]:
x[2, 2] := x[1, 2] + a[1] ∗ p[1, 2]:
  for i for 1 to 2 do
x[2, i] := x[1, i] + a[1, i]
  end for:
c[2, 1] := (f (x[1, 1] + rho ∗ u[2, 1] − sigma ∗ p[1, 1], x[1, 2] + rho ∗ u[2, 2] − sigma ∗ p[1, 2]) − f (x[1, 1] + rho ∗
u[2, 1] + sigma ∗ p[1, 1], x[1, 2] + rho ∗ u[2, 2] + sigma ∗ p[1, 2]))/(2 ∗ sigma):
a[2, 1] := c[2, 1]/d[1]:
b[2, 1] := (a[2, 1] − a[1])/rho:
  for i from 1 to 2 do
pbar[2, i] := u[2, i] + b[2, 1] ∗ p[1, i]
  end for:
  for i from 1 to 2 do
p[2, i] := pbar[2, i]/sqrt((pbar[2, 1])2 + (pbar[2, 2])2)
  end for:
  for k from 2 to 2 do
c[k] := (f (x[1, 1] − sigma ∗ p[k, 1], x[1, 2] − sigma ∗ p[k, 2]) − f (x[1, 1] + sigma ∗ p[k, 1], x[1, 2] + sigma ∗
p[k, 2]))/(2 ∗ sigma)
  end for:
  for k from 2 to 2 do
d[k] := (f (x[1, 1] − sigma ∗ p[k, 1], x[1, 2] − sigma ∗ p[k, 2]) − 2 ∗ f (x[1, 1], x[1, 2]) + f (x[1, 1] + sigma ∗
p[k, 1], x[1, 2] + sigma ∗ p[k, 2]))/(sigma)2
  end for:
  for k from 2 to 2 do
a[k] := c[k]/d[k]
  end for;
  for i from 1 to 2 do
x[3, i] := x[2, i] + a[2] ∗ p[2, i]
  end for;
  for i from 1 to 2 do
x[1, i] := x[3, i]
  end for;
print (x[3, 1], x[3, 2]);
y[ j, 1] := x[3, 1]; y[ j, 2] := x[3, 2];
end for

j.ijmce-2025-0010.tab.006

for i from 1 to 9 do
  C[i] = sqrt((y[i + 1, 1] − 1)2 + (y[i + 1, 2] − 1)2)/(sqrt((y[i, 1] − 1)2 + (y[i, 2] − 1)2));
end for;

The asymptotic constants of a nonlinear function.

Asymptotic Constant CGS method Newton method Steepest Descent method Runge-Kutta method
C1 0.857485 0.857485 0.453589 0.785244
C2 0.027425 0.027425 0.454544 0.785678
C3 0.243358 0.243358 0.455502 0.786108
C4 0.003072 0.003072 0.456463 0.786534
C5 0.200003 0.200003 0.457428 0.786955
C6 0.003073 0.003073 0.458396 0.787372
C7 0.200002 0.200002 0.459367 0.787783
C8 0.003073 0.003073 0.460341 0.788190
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