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## NLPNMS Call

nonlinear optimization by Nelder-Mead simplex method

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

See the section "Nonlinear Optimization and Related Subroutines" for a listing of all NLP subroutines. See Chapter 11 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 can be unable to generate precise results when .

The NLPNMS subroutine uses the following simplex algorithms:

• For unconstrained or only boundary-constrained problems, the original Nelder-Mead simplex algorithm is implemented and extended to boundary constraints. This algorithm does not compute the objective for infeasible points, and it is invoked if the "nlc" module argument is not specified and the blc argument contains at most two rows (corresponding to lower and upper bounds).
• For linearly or nonlinearly constrained problems, a slightly modified version of Powell's (1992) Constrained Optimization BY Linear Approximations (COBYLA) implementation is used. This algorithm is invoked if the "nlc" module argument is specified or if at least one linear constraint is specified with the blc argument.

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, , of a spheric trust region. The modification implemented in the NLPNMS call permits an increase of the trust-region radius in special situations. A sequence of iterations is performed with a constant trust-region radius until the computed function reduction is much less than the predicted reduction. Then, the trust-region radius 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, , can be specified with the second element of the par argument, and the final radius, , can be specified with the ninth element of the tc argument. Convergence to small values of , or high-precision convergence, can require many calls of the function and constraint modules and can result in numerical problems. The main reasons for the slow convergence of the COBYLA algorithm are as follows:

• Linear approximations of the objective and constraint functions are used locally.
• Maintaining the regularly shaped simplex and not adapting its shape to nonlinearities yields very small simplexes for highly nonlinear functions, such as fourth-order polynomials.
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 the section "Parameter Constraints" for details.

For more information about the special sets of termination criteria used by the NLPNMS algorithms, see the section "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

• difcrit, which, in this subroutine, refers to the difference between the largest and smallest function values of the simplex vertices
• std, which is the standard deviation of the function values of the simplex vertices
• deltax, which is the vertex length of a restarted simplex. If there are convergence problems, the algorithm restarts the iteration process with a simplex of smaller vertex length.
• size, which is the average distance of the simplex vertex with the smallest function value to the other simplex vertices
For linearly and nonlinearly constrained problems, the subroutine prints the following:
• conmax is the maximum constraint violation.
• meritf is the value of the merit function, .
• difmerit is the difference between adjacent values of the merit function.
• is the trust-region radius.

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


start F_HS43(x);
f = x*x + x[3]*x[3] - 5*(x[1] + x[2]) - 21*x[3] + 7*x[4];
return(f);
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];
return(c);
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 follows.



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

Maximum
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

Between
Actual
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    *?*
`

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