Optimization: Algorithms and Applications

1. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   
2. % MATLAB code bisection.m
   
3. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   
4. %
   
5. % a-> lower bound of the design variable
   
6. % b> upper bound of the design variable
   
7. % alpha -> midpoint of a and b
   
8. % delx -> ?x for central difference method
   
9. % derivative -> derivative using central difference method
   
10. % derivative_alpha -> derivative at x = alpha
    
11. % abs -> absolute of a number, MATLAB function
    
12. %
    
13. clear all
    
14. clc
    
15. a = 40;
    
16. b = 90;
    
17. epsilon = 0.01;
    
18. delx = 0.01;
    
19. fprintf('    a             b     \n' )
    
20. fprintf('-------------------------\n' )
    
21. 
    
22. for i= 1:100
    
23. fprintf(' %7.3f      %8.3f \n',a,b)
    
24. alpha = (a+b)/2;
    
25. derivative = (func(a+delx) - func(a-delx) )/(2*delx);
    
26. derivative_alpha = (func(alpha+delx)- func(alpha-delx))/(2*delx);
    
27. if (derivative*derivative_alpha) < 0
    
28.                     b = alpha;
    
29. else
    
30.             a = alpha;
    
31. end
    
32. 
    
33. if abs(a-b) < epsilon
    
34.                     break;
    
35. end
    
36. end
    
37. 
    
38. fprintf('-------------------------\n' )
    
39. fprintf('x* =  %7.3f       Minimum =   %8.3f\n',a,func(a))
    
40. fprintf('Number of function calls =     %3d\n',4*i)
    
41. %
    
42. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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1. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   
2. % MATLAB code cubic
   
3. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   
4. %
   
5. % a-> lower bound of the design variable
   
6. % b> upper bound of the design variable
   
7. % alpha -> midpoint of a and b
   
8. % delx -> ?x for central difference method
   
9. % derivative -> derivative using central difference method
   
10. % derivative_alpha -> derivative at x = alpha
    
11. % abs -> absolute of a number, MATLAB function
    
12. % flag -> set the flag when minimum is bracketed
    
13. % derivative_a -> derivative at point a
    
14. % derivative_b -> derivative at b
    
15. %
    
16. a = 40;
    
17. b = 90;
    
18. delx = 0.01;
    
19. flag = 0;
    
20. epsilon= 0.001;
    
21. fprintf('    a             b     \n' )
    
22. fprintf('-------------------------\n' )
    
23. 
    
24. for i= 1:100
    
25. alpha = (a+b)/2;
    
26. derivative = (func(a+delx) - func(a-delx) )/(2*delx);
    
27. derivative_alpha = (func(alpha+delx)-func(alpha-delx) )/(2*delx);
    
28. if (derivative*derivative_alpha) < 0
    
29.     b = alpha;
    
30.     flag = 1;
    
31. else
    
32.     a = alpha;
    
33. end
    
34. if flag == 1
    
35.     break;
    
36. end
    
37. end
    
38. 
    
39. for j=1:100
    
40. fprintf(' %7.3f      %8.3f \n',a,b)
    
41. derivative_a = (func(a+delx) - func(a-delx) )/(2*delx);
    
42. derivative_b = (func(b+delx) - func(b-delx) )/(2*delx);
    
43. z = 3*(func(a)-func(b))/(b-a) + derivative_a + derivative_b;
    
44. w = ((b-a)/abs(b-a))*sqrt(z*z-derivative_a*derivative_b);
    
45. mew = (derivative_b+w-z)/(derivative_b-derivative_a+2*w);
    
46. if mew <= 1
    
47.     x_opt = b - mew*(b-a);
    
48. else
    
49.     x_opt = a;
    
50. end
    
51. alpha1 = (func(x_opt+delx) - func(x_opt-delx) )/(2*delx);
    
52. if abs(alpha1) < epsilon
    
53.     break;
    
54. else
    
55.    if (derivative_a*alpha1) < 0
    
56.     b = x_opt;
    
57. else
    
58.     a = x_opt;
    
59. end
    
60. end
    
61. 
    
62. end
    
63. 
    
64. fprintf('-------------------------\n' )
    
65. fprintf('x* =  %7.3f       Minimum =   %8.3f\n',x_opt,func(x_opt))
    
66. fprintf('Number of function calls =     %3d\n',4*i+8*j)
    
67. %
    
68. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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1. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   
2. % MATLAB code golden.m
   
3. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   
4. %
   
5. % a-> lower bound of the design variable
   
6. % b> upper bound of the design variable
   
7. % alpha -> midpoint of a and b
   
8. % falpha1 -> function value at x = alpha1
   
9. % falpha2 -> function value at x = alpha2
   
10. % epsilon -> constant used to terminate the algorithm
    
11. % abs -> absolute of a number, MATLAB function
    
12. % tau -> 2-golden number
    
13. %
    
14. clear all
    
15. clc
    
16. a = 40;
    
17. b = 90;
    
18. epsilon = 0.00001;
    
19. tau = 0.381967;
    
20. alpha1 = a*(1-tau) + b*tau;
    
21. alpha2 = a*tau + b*(1-tau);
    
22. falpha1 = func(alpha1);
    
23. falpha2 = func(alpha2);
    
24. 
    
25. fprintf('    a             b     \n' )
    
26. fprintf('-------------------------\n' )
    
27. 
    
28. for i= 1:100
    
29.     fprintf(' %7.3f      %8.3f \n',a,b)
    
30.     if falpha1 > falpha2
    
31.         a = alpha1;
    
32.         alpha1 = alpha2;
    
33.         falpha1 = falpha2;
    
34.         alpha2 = tau*a + (1-tau)*b;
    
35.         falpha2 = func(alpha2);
    
36.     else
    
37.         b = alpha2;
    
38.         alpha2 = alpha1;
    
39.         falpha2 = falpha1;
    
40.         alpha1 = tau*b + (1-tau)*a;
    
41.         falpha1 = func(alpha1);
    
42.     end
    
43. 
    
44. if abs(func(alpha1)-func(alpha2))< epsilon
    
45.     break;
    
46. end
    
47. end
    
48. 
    
49. fprintf('-------------------------\n' )
    
50. fprintf('x* =  %7.3f       Minimum =   %8.3f\n',alpha1,func(alpha1))
    
51. fprintf('Number of function calls =     %3d\n',2+i)
    
52. %
    
53. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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1. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   
2. % MATLAB code newtonraphson.m
   
3. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   
4. %
   
5. % x -> initial guess of design variable
   
6. % delx -> ?x for central difference method
   
7. % derivative -> derivative using central difference method
   
8. % sec_derivative -> second derivative
   
9. % epsilon -> constant used to terminate the program
   
10. % xprev -> value of x stored from previous iteration
    
11. %
    
12. clear all
    
13. clc
    
14. x = 45;
    
15. delx = 0.01;
    
16. epsilon = 0.01;
    
17. fprintf('     x    f(x)      Deriv. Second deriv.\n' )
    
18. fprintf('-----------------------------------------\n' )
    
19. 
    
20. for i= 1:100
    
21. derivative = (func(x+delx) - func(x-delx) )/(2*delx);
    
22. sec_derivative=(func(x+delx)+func(x-delx)-2*func(x))/(delx*delx);
    
23. fprintf('%8.3f %8.3f %8.3f %8.3f\n',x,func(x),derivative,      sec_derivative)
    
24. xprev = x;
    
25. x = x- derivative/sec_derivative;
    
26. if abs(x-xprev) < epsilon
    
27.     break;
    
28. end
    
29. end
    
30. 
    
31. fprintf('-----------------------------------------\n' )
    
32. fprintf('Number of function calls =     %3d\n',5*i)
    
33. 
    
34. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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1. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   
2. % MATLAB code secant.m
   
3. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   
4. %
   
5. % a-> lower bound of the design variable
   
6. % b> upper bound of the design variable
   
7. % alpha -> midpoint of a and b
   
8. % delx -> ?x for central difference method
   
9. % derivative -> derivative using central difference method
   
10. % derivative_alpha -> derivative at x = alpha
    
11. % abs -> absolute of a number, MATLAB function
    
12. % flag -> set the flag when minimum is bracketed
    
13. %
    
14. clear all
    
15. clc
    
16. a = 40;
    
17. b = 90;
    
18. epsilon = 0.001;
    
19. delx = 0.01;
    
20. flag = 0;
    
21. fprintf('  Alpha          Deriv.  \n' )
    
22. fprintf('-------------------------\n' )
    
23. 
    
24. for i= 1:100
    
25. alpha = (a+b)/2;
    
26. derivative = (func(a+delx) - func(a-delx) )/(2*delx);
    
27. derivative_alpha = (func(alpha+delx)-func(alpha-delx) )/(2*delx);
    
28. if (derivative*derivative_alpha) < 0
    
29.     b = alpha;
    
30.     flag = 1;
    
31. else
    
32.     a = alpha;
    
33. end
    
34. if flag == 1
    
35.     break;
    
36. end
    
37. end
    
38. 
    
39. for  j = 1:100
    
40. fprintf(' %7.3f      %8.3f \n',alpha,derivative_alpha)
    
41. derivative_a = (func(a+delx) - func(a-delx) )/(2*delx);
    
42. derivative_b = (func(b+delx) - func(b-delx) )/(2*delx);
    
43. alpha = b - derivative_b*(b-a)/(derivative_b-derivative_a);
    
44. derivative_alpha =(func(alpha+delx) -func(alpha-delx) )/(2*delx);
    
45. if derivative_alpha > 0
    
46.     b = alpha;
    
47. else
    
48.     a = alpha;
    
49. end
    
50. if abs(derivative_alpha) < epsilon
    
51.     break;
    
52. end
    
53. end
    
54. fprintf('-------------------------\n' )
    
55. fprintf('x* =  %7.3f       Minimum =   %8.3f\n',alpha,func(alpha))
    
56. fprintf('Number of function calls =     %3d\n',4*i+6*j)
    
57. %
    
58. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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