求:关于序列二次规划的详细描述

fea_stud

大家有关于序列二次规划的详细描述的资料吗?
或者在什么书上有?

光的轨迹

在isight的user reference里有描述，大致的算法介绍和流程都有的。
优化算法书上也大都有对二次规划的讲解。

1. 
   
2. SQP Method - Short summary
   
3. Notation: vectors are written in bold, e.g. x represents the vector of design variables.
   
4. 
   
5. Optimization problem:
   
6. 
   
7. Min f(x)
   
8. 
   
9. Subject to:
   
10. 
    
11. h(x) = 0 (equality constraints)
    
12. 
    
13. g(x) £ 0 (inequality constraints)
    
14. 
    
15. Necessary condition for point x* to be a minimizer = KKT condition (Karush-Kuhn-Tucker condition):
    
16. 
    
17. Ñf* + lTÑh* + mTÑg = 0T
    
18. 
    
19. with
    
20. 
    
21. l ¹ 0
    
22. 
    
23. mTg = 0, m ³ 0
    
24. 
    
25. l and m are the Lagrange multipliers.
    
26. 
    
27. An inequality constraint can be inactive, in that case gj < 0, mj = 0.
    
28. 
    
29. If the inequality constraint is active, gj = 0, mj &sup1; 0, and it can be handled as an equality constraint.
    
30. 
    
31. Sufficiency condition for point x* to be a minimizer: Hessian matrix (second derivative) should be positive definite: H > 0
    
32. 
    
33. SQP algorithm:
    
34. 
    
35. -        Select starting point x0, initialize l0, initialize H, let k (iteration counter) = 0
    
36. 
    
37. -        First gradient calculation
    
38. 
    
39. -        Iterate while convergence criteria not met:
    
40. 
    
41. -  if k &sup1; 0, update Hessian matrix using the BFGS method
    
42. 
    
43. -  Solve quadratic subproblem with linearized constraints following Newton’s method. This results in:
    
44. 
    
45. sk, the search direction
    
46. 
    
47. lk+1, the lagrange multipliers
    
48. 
    
49. -  Minimize a merit function along sk, to determine a step length ?k
    
50. 
    
51. -  Set xk+1 = xk + aksk
    
52. 
    
53. -  Calculate gradient for xk+1 and check convergence (KKT condition)
    
54. 
    
55. -  Let k = k+1
    
56. 
    
57. The merit function is the one proposed by Powell:
    
58. 
    
59. F(x,l,&micro;) = f(x) + Swi|hi| + Swj|min{0,-gj}|
    
60. 
    
61. wi and wj are the weights that are used to balance the infeasibilities. The values for the weights are taken as follows:
    
62. 
    
63. wi = |li| for k = 0 (first iteration)
    
64. 
    
65. wi,k = max{|li,k|,0.5(wi,k-1-1+|li,k|)} for k > 0
    
66. 
    
67. Similar formulas using mj are used for the weights for the inequalities.
    
68. 
    
69. Design variables are always normalized to the interval [-1,1] to give equal weight to different dynamic ranges.
    
70. 
    
71. Constraint values are always normalized by dividing by a range, which is |H-L| with H the high constraint value and L the low constraint value. If only one of both (L or H) is specified, the other is taken such that H = 1.2L if H > 0 and H = L/1.2 if H < 0.
    
72. 
    
73. The merit function is always a compromise between minimizing f(x) and satisfying the constraints, hence it is possible that a better point is found while constraints are not satisfied, because the total merit function is better. The merit function can be influenced by multiplying the objective function with a factor, which results in a bigger or smaller weight in the merit function.
    

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光的轨迹

怎么贴出来这么多的鬼脸呢！

ylai

是不是( )是表情符？