Mnfb Mnfb_r Mngb Mngb_r Mnhb Mnhb_r

Mnfb_r() Sub Mnfb_r ( N As  Long, X() As  Double, B() As  Double, Info As  Long, YY As  Double, IRev As  Long, Optional Iout As  Long = 0, 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) (reverse communication version) 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. The user need not supply the analytic gradient values. The secant method (BFGS update) is used to compute the Hessian. Steps are computed by the double dogleg trust region method. Mnfb_r is the reverse communication version of Mnfb. 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] IRev = 0: Obtained solution vector.   IRev = 1: The abscissa where the function value shoule be evaluated.   IRev = 3: Recent approximation of the 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) [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,out] YY [in] If IRev = 1, the function value at X() should be given in YY in the next call. [out] If IRev = 3, YY contains the function value at current X() for printing. [in,out] IRev Control variable for reverse communication. [in] Before first call, IRev should be initialized to zero. On succeeding calls, IRev should not be altered. [out] If IRev is not zero, set the required values to the specified variables or display the intermediate result as follows and call the routine again. = 0: Computation finished. See return code in Info. = 1: User should set the function value at X() in YY. Do not alter any variables other than YY. = 3: To be returned with IRev = 3 on each iteration if Iout = 1. Output the intermediate result (X(), NFcall, NGcall, Niter, Fval, etc.). Do not alter any variables. [in] Iout (Optional) Whether the intermediate result output is necessary. (default = 0) = 0: Output is not necessary (never return with IRev = 3). = 1: Return on each iteration with IRev = 3 to output the intermediate results. (If other value is specified , Iout = 0 will be assumed) [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 with IRev = 1 (excluding those for computing gradients). [out] NGcall (Optional) Number of function evaluations with IRev = 1 (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 Ex_Mnfb_r() Const N As Long = 2 Dim X(N - 1) As Double, B(1, N - 1) As Double, Info As Long Dim YY As Double, IRev 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 IRev = 0 Do Call Mnfb_r(N, X(), B(), Info, YY, IRev) If IRev = 1 Then YY = 100 * (X(1) - X(0) ^ 2) ^ 2 + (1 - X(0)) ^ 2 End If Loop While IRev <> 0 Debug.Print "X1, X2 =", X(0), X(1) Debug.Print "Info =", Info End Sub Example Results X1, X2 = 1.00000000015727 1.00000000031223 Info = 0