Mnfb Mnfb_r Mngb Mngb_r Mnhb Mnhb_r

Mnfb() Sub Mnfb ( N As  Long, X() As  Double, B() As  Double, F As  LongPtr, Info As  Long, Optional Itsum As  LongPtr = NullPtr, Optional Info2 As  Long, Optional NFcall As  Long, Optional NGcall As  Long, Optional Niter As  Long, Optional Fval As  Double, Optional Rtol As  Double = -1, Optional Atol As  Double = -1, Optional MaxFcall As  Long = 0, Optional MaxIter As  Long = 0, Optional Tuner1 As  Double = -1, Optional Xctol As  Double = -1, Optional Xftol As  Double = -1, Optional Lmax0 As  Double = -1, Optional Lmaxs As  Double = -1, Optional Sctol As  Double = -1, Optional Bias As  Double = -1, Optional Eta0 As  Double = -1  ) Minimum of a multivariable nonlinear function (trust region method) (simply bounded) (gradient computed by finite differences) Purpose This routine finds the local minimum point (xs1, xs2, ..., xsn) of general nonlinear function f(x1, x2, ..., xn) (a twice continuously differentiable real-valued function) under the following simple constraint (b0i and b1i are lower and upper bounds). b0i <= xsi <= b1i (i = 1 to n) The gradient of the objective function is calculated by the finite differences instead of calling user-supplied analytical routine. The secant method (BFGS update) is used to compute the Hessian. Steps are computed by the double dogleg trust region method. Parameters [in] N The number of variables. (N > 0) [in,out] X() Array X(LX - 1) (LX >= N) [in] Initial approximation of the solution vector. [out] Obtained solution vector. [in] B() Array B(LB1 - 1, LB2 - 1) (LB1 = 2, LB2 >= N) Lower and upper bounds on solution vector X.   B(0, I) <= X(I) <= B(1, I) (I = 0 to N-1) [in] F The user-supplied subroutine which calculates the function f(x1, x2, ..., xn) defined as follows. Sub F(N As Long, X() As Double, Nf As Long, Fval As Double) Calculate the function value from given N and X() and return in Fval. Nf is invocation counter. If given X() is out of bounds, Nf should be set to 0. The other variables should not be altered. End Sub [out] Info = 0: Successful exit. (See sub-code in Info2) = -1: The argument N had an illegal value. (N < 1) = -2: The argument X() is invalid. = -3: The argument B() is invalid. = 7: Singular convergence. (Hessian near the current iterate appears to be singular) = 8: False convergence. (Iterate appears to be converging to a noncritical point. Tolerances may be too small) = 9: Function evaluation limit reached. = 10: Iteration limit reached. = 63: F(X) cannot be computed at the initial X. = 65: The gradient could not be computed at X. [in] Itsum (Optional) The user supplied subroutine to print the intermediate results defined as follows. (default = NUllPtr) If the address is supplied (if Itsum <> NullPtr), the subroutine is called after every iteration. Sub Itsum(N As Long, X() As Double, NIter As Long, Nf As Long, Ng As Long, Fval As Double) Output the following information in desired format. N: Number of variables. X(): Current approximation of the solution vector. NIter: Iteration counter. Nf: Number of function calls of F (excluding those for computing gradients). Ng: Number of function calls of F (for computing gradients). Fval: F value at X(). End Sub Argument values should not be altered. [out] Info2 (Optional) Sub-code for Info = 0. = 1: X convergence. = 2: Relative function convergence. = 3: Both X and relative function convergence. = 4: Absolute function convergence. [out] NFcall (Optional) Number of function evaluations of F (excluding those for computing gradients). [out] NGcall (Optional) Number of function evaluations of F (for computing gradients). [out] Niter (Optional) Number of iterations. [out] Fval (Optional) Function value at the obtained solution vector X(). [in] Rtol (Optional) Relative function convergence tolerance. (Eps <= Rtol <= 0.1) (default = 1e-10) (Eps: machine epsilon) (If Rtol < Eps or Rtol > 0.1, the default value will be used) [in] Atol (Optional) Absolute function convergence tolerance. (default = 1e-20) (If Atol < 0, the default value will be used) [in] MaxFcall (Optional) Maximum number of function evaluations of F. (default = 200) (If MaxFcall <= 0, the default value will be used) [in] MaxIter (Optional) Maximum number of iterations. (default = 150) (If MaxIter <= 0, the default value will be used) [in] Tuner1 (Optional) Parameter to check for false convergence. (0 <= Tuner1 <= 0.5) (default = 0.1) (If Tuner1 < 0 or Tuner1 > 0.5, the default value will be used) [in] Xctol (Optional) X convergence tolerance. (0 <= Xctol <= 1) (default = Eps^(1/2)) (If Xctol < 0 or Xctol > 1, the default value will be used) [in] Xftol (Optional) False convergence tolerance. (0 <= Xftol <= 1) (default = 100*Eps) (If Xftol < 0 or Xftol > 1, the default value will be used) [in] Lmax0 (Optional) Maximum 2-norm allowed for scaled very first step. (Lmax0 > 0) (default = 1) (If Lmax0 <= 0, the default value will be used) [in] Lmaxs (Optional) [in] Sctol (Optional) Lmaxs and Sctol are the singular convergence test parameters. (Lmaxs > 0, 0 <= Sctol <= 1) (default: Lmaxs = 1, Sctol = 1e-10) To test if the function reduction predicted for a step of length bounded by Lmaxs is at most Sctol*abs(f) (f is the function value at the start of the current iteration). (If Lmaxs <= 0, the default value will be used) (If Sctol < 0 or Sctol > 1, the default value will be used) [in] Bias (Optional) The bias parameter used in the dogleg trust region method. (0 <= Bias <= 1) (default = 0.8) (If Bias < 0 or Bias > 1, the default value will be used) [in] Eta0 (Optional) Estimated bound on the relative error in the function value computed by F (suppose (true value) = (computed value)*(1 + E), Abs(E) <= Eta0). (Eps <= Eta0 <= 1) (default = 1000*Eps) (Eps: machine epsilon) (If Eta0 <= 0, the default value will be used) Reference netlib/port Example Program Find the minimum point of the following function (Rosenbrock function). f(x1, x2) = 100(x2 - x1^2)^2 + (1 - x1)^2 The initial approximation is (x1, x2) = (-1.2, 1). Lower and upper bounds on solution are -2 <= x1 <= 2 and 0 <= x2 <= 2. Sub FMnfb(N As Long, X() As Double, Nf As Long, F As Double) F = 100 * (X(1) - X(0) ^ 2) ^ 2 + (1 - X(0)) ^ 2 End Sub Sub Ex_Mnfb() Const N As Long = 2 Dim X(N - 1) As Double, B(1, N - 1) As Double, Info As Long X(0) = -1.2: X(1) = 1 B(0, 0) = -2: B(1, 0) = 2 B(0, 1) = 0: B(1, 1) = 2 Call Mnfb(N, X(), B(), AddressOf FMnfb, Info) Debug.Print "X1, X2 =", X(0), X(1) Debug.Print "Info =", Info End Sub Example Results X1, X2 = 1.00000000015727 1.00000000031223 Info = 0