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nonlinear optimization by Nelder-Mead simplex method

CALL NLPNMS( rc, xr, "fun", x0 <,opt, blc, tc, par, "ptit", "nlc">);

See "Nonlinear Optimization and Related Subroutines" for a listing of all NLP subroutines. See Chapter 11, "Nonlinear Optimization Examples," for a description of the inputs to and outputs of all NLP subroutines.

The Nelder-Mead simplex method is one of the subroutines that can solve optimization problems with nonlinear constraints. It does not use any derivatives, and it does not assume that the objective function has continuous derivatives. However, the objective function must be continuous. The NLPNMS technique uses a large number of function calls, and it may be unable to generate precise results when n>40.

The NLPNMS subroutine uses the following simplex algorithms:

The original Nelder-Mead algorithm cannot be used for general linear or nonlinear constraints, but in the unconstrained or boundary-constrained cases, it can be faster. It changes the shape of the simplex by adapting the nonlinearities of the objective function; this contributes to an increased speed of convergence.

Powell's COBYLA Algorithm

Powell's COBYLA algorithm is a sequential trust-region algorithm that tries to maintain a regularly-shaped simplex throughout the iterations. The algorithm uses a monotone-decreasing radius, \rho, of a spheric trust region. The modification implemented in the NLPNMS call permits an increase of the trust-region radius \rho in special situations. A sequence of iterations is performed with a constant trust-region radius \rho until the computed function reduction is much less than the predicted reduction. Then, the trust-region radius \rho is reduced. The trust-region radius is increased only if the computed function reduction is relatively close to the predicted reduction and if the simplex is well-shaped. The start radius, \rho_{beg}, can be specified with the second element of the par argument, and the final radius, \rho_{end}, can be specified with the ninth element of the tc argument. Convergence to small values of \rho_{end}, or high-precision convergence, may require many calls of the function and constraint modules and may result in numerical problems. The main reasons for the slow convergence of the COBYLA algorithm are as follows: To allocate memory for the vector returned by the "nlc" module argument, you must specify the total number of nonlinear constraints with the tenth element of the opt argument. If any of the constraints are equality constraints, the number of equality constraints must be specified by the eleventh element of the opt argument. See "Parameter Constraints" for details.

For more information on the special sets of termination criteria used by the NLPNMS algorithms, see "Termination Criteria"

In addition to the standard iteration history, the NLPNMS subroutine prints the following information. For unconstrained or boundary-constrained problems, the subroutine also prints

For linearly and nonlinearly constrained problems, the subroutine prints the following:

The following code uses the NLPNMS call to solve the Rosen-Suzuki problem (see "Rosen-Suzuki Problem" ), which has three nonlinear constraints:

   proc iml;
      start F_HS43(x);
         f = x*x` + x[3]*x[3] - 5*(x[1] + x[2]) - 21*x[3] + 7*x[4];
      finish F_HS43;
      start C_HS43(x);
         c = j(3,1,0.);
         c[1] = 8 - x*x` - x[1] + x[2] - x[3] + x[4];
         c[2] = 10 - x*x` - x[2]*x[2] - x[4]*x[4] + x[1] + x[4];
         c[3] = 5 - 2.*x[1]*x[1] - x[2]*x[2] - x[3]*x[3]
                  - 2.*x[1] + x[2] + x[4];
      finish C_HS43;
      x = j(1,4,1);
      optn= j(1,11,.); optn[2]= 3; optn[10]= 3; optn[11]=0;
      call nlpnms(rc,xres,"F_HS43",x,optn,,,,,"C_HS43");

Part of the output produced by the preceding code is shown in Figure 17.6.

                Optimization Start
                Parameter Estimates
           N Parameter         Estimate

           1 X1                1.000000
           2 X2                1.000000
           3 X3                1.000000
           4 X4                1.000000

        Value of Objective Function = -19

                Values of Nonlinear Constraints

           Constraint         Residual

           [         1 ]      4.0000
           [         2 ]      6.0000
           [         3 ]      1.0000

         Nelder-Mead Simplex Optimization

      COBYLA Algorithm by M.J.D. Powell (1992)

  Minimum Iterations                                     0
  Maximum Iterations                                  1000
  Maximum Function Calls                              3000
  Iterations Reducing Constraint Violation               0
  ABSFCONV Function Criterion                            0
  FCONV Function Criterion                    2.220446E-16
  FCONV2 Function Criterion                           1E-6
  FSIZE Parameter                                        0
  ABSXCONV Parameter Change Criterion               0.0001
  XCONV Parameter Change Criterion                       0
  XSIZE Parameter                                        0
  ABSCONV Function Criterion                  -1.34078E154
  Initial Simplex Size (INSTEP)                        0.5
  Singularity Tolerance (SINGULAR)                    1E-8
        Nelder-Mead Simplex Optimization

    COBYLA Algorithm by M.J.D. Powell (1992)

                Parameter Estimates                 4
                Nonlinear Constraints               3

               Optimization Start

Objective Function   -29.5  Maximum Constraint Violation   4.5

                         Function       Objective    Constraint      
    Iter     Restarts       Calls        Function     Violation    

       1            0          12       -52.80342        4.3411   
       2            0          17       -39.51475        0.0227   
       3            0          53       -44.02098       0.00949  
       4            0          62       -44.00214      0.000833   
       5            0          72       -44.00009      0.000033   
       6            0          79       -44.00000      1.783E-6  
       7            0          90       -44.00000      1.363E-7  
       8            0          94       -44.00000      1.543E-8  

                            Merit          and
                Merit    Function    Predicted
    Iter     Function      Change       Change

       1     -42.3031      12.803        1.000
       2     -39.3797      -2.923        0.250
       3     -43.9727       4.593       0.0625
       4     -43.9977      0.0249       0.0156
       5     -43.9999     0.00226       0.0039
       6     -44.0000     0.00007       0.0010
       7     -44.0000     1.74E-6       0.0002
       8     -44.0000     5.33E-7       0.0001

                                          Optimization Results

Iterations                              8  Function Calls                95
Restarts                                0  Objective Function  -44.00000003
Maximum Constraint Violation  1.543059E-8  Merit Function      -43.99999999
Actual Over Pred Change            0.0001

   ABSXCONV convergence criterion satisfied.

WARNING: The point x is feasible only at the LCEPSILON= 1E-7 range.

             Optimization Results
             Parameter Estimates
       N Parameter         Estimate

       1 X1            -0.000034167
       2 X2                1.000004
       3 X3                2.000023
       4 X4               -0.999971

 Value of Objective Function = -44.00000003

         Values of Nonlinear Constraints

          Constraint       Residual

          [         1 ]    -1.54E-8    *?*
          [         2 ]      1.0000
          [         3 ]     -1.5E-8    *?*

Figure 17.6: Iteration History for the NLPNMS Subroutine

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