specifies the method for computing the generalized (G2 or G4) inverse of a singular matrix needed for the approximate covariance matrix of parameter estimates. This option is valid only for applications where the approximate covariance matrix of parameter estimates is found to be singular.

specifies the relative function convergence criterion. For more details, see the section Termination Criteria Options.

specifies the relative gradient convergence criterion. For more details, see the section Termination Criteria Options.

restricts the step length of an optimization algorithm during the first iterations.

specifies the line-search method for the CONGRA, QUANEW, and NEWRAP optimization techniques.

specifies the degree of accuracy that should be obtained by the line-search algorithms LIS=2 and LIS=3.

specifies the maximum number of function calls in the optimization process. For more details, see the section Termination Criteria Options.

specifies the maximum number of iterations in the optimization process. For more details, see the section Termination Criteria Options.

specifies the optimization technique.

specifies the update method for the quasi-Newton or conjugate-gradient optimization technique.

specifies an absolute singularity criterion , , for the inversion of the information matrix, which is needed to compute the approximate covariance matrix of parameter estimates.

specifies a nonnegative threshold , , that decides whether the eigenvalues of the information matrix are considered to be zero. This option is valid only for applications where the approximate covariance matrix of parameter estimates is found to be singular.

specifies a relative singularity criterion , , for the inversion of the information matrix, which is needed to compute the approximate covariance matrix of parameter estimates.

specifies the singularity criterion , , that is used for the inversion of the Hessian matrix. The default value is 1E8.

specifies a relative singularity criterion , , for the inversion of the information matrix, which is needed to compute the approximate covariance matrix of parameter estimates.

specifies an absolute function convergence criterion.

• For minimization, termination requires

• For maximization, termination requires

The default value of ABSCONV is as follows:

• for minimization, the negative square root of the largest double-precision value

• for maximization, the positive square root of the largest double-precision value

requires the number of function calls to be no larger than . The default values are listed in the following table.

The default is used if you specify MAXFUNC=0. The optimization can be terminated only after completing a full iteration. Therefore, the number of function calls that are actually performed can exceed the number that is specified by the MAXFUNC= option.

requires the number of iterations to be no larger than . The default values are listed in the following table.

The default is used if you specify MAXITER=0 or you omit the MAXITER option.

The optional second value is valid only for TECH=QUANEW with nonlinear constraints. It specifies an upper bound for the number of iterations of an algorithm and reduces the violation of nonlinear constraints at a starting point. The default value is =20. For example, specifying MAXITER= . 0 means that you do not want to exceed the default number of iterations during the main optimization process and that you want to suppress the feasible-point algorithm for nonlinear constraints.

requires the CPU time to be no larger than . The default value of the MAXTIME= option is the largest double floating-point number on your computer.

specifies the minimum number of iterations. The default value is .

The ABSCONV=, MAXITER=, MAXFUNC=, and MAXTIME= options are useful for dividing a time-consuming optimization problem into a series of smaller problems by using the OUTEST= and INEST= data sets.

specifies the absolute function convergence criterion. Termination requires a small change of the function value in successive iterations,

The default value is . The optional integer value determines the number of successive iterations for which the criterion must be satisfied before the process can be terminated.

specifies the absolute gradient convergence criterion. Termination requires the maximum absolute gradient element to be small,

The default value is =1E5. The optional integer value determines the number of successive iterations for which the criterion must be satisfied before the process can be terminated.

Note:In some applications, the small default value of the ABSGCONV= criterion is too difficult to satisfy for some of the optimization techniques.

specifies the absolute parameter convergence criterion. Termination requires a small Euclidean distance between successive parameter vectors,

The default value is . The optional integer value determines the number of successive iterations for which the criterion must be satisfied before the process can be terminated.

specifies the relative function convergence criterion. Termination requires a small relative change of the function value in successive iterations,

where is defined by the FSIZE= option. The default value is , where either is specified or is set by default to , where is the machine precision. The optional integer value determines the number of successive iterations for which the criterion must be satisfied before the process can be terminated.

specifies another function convergence criterion. For least squares problems, termination requires a small predicted reduction

of the objective function.

The predicted reduction

is computed by approximating the objective function by the first two terms of the Taylor series and substituting the Newton step

The FCONV2 criterion is the unscaled version of the GCONV criterion. The default value is . The optional integer value determines the number of successive iterations for which the criterion must be satisfied before the process can be terminated.

specifies the number of accurate digits in evaluations of the objective function. Fractional values such as FDIGITS=4.7 are allowed. The default value is , where is the machine precision. The value of is used for the specification of the default value of the FCONV= option.

specifies the parameter of the relative function and relative gradient termination criteria. The default value is . See the FCONV= and GCONV= options for details.

specifies the relative gradient convergence criterion. For all techniques except the CONGRA technique, termination requires that the normalized predicted function reduction be small,

where is defined by the FSIZE= option. For the CONGRA technique (where a reliable Hessian estimate is not available),

is used. The default value is =1E8. The optional integer value determines the number of successive iterations for which the criterion must be satisfied before the process can be terminated.

Note:The default setting for the GCONV= option sometimes leads to early termination far from the location of the optimum. This is especially true for the special form of this criterion used in the CONGRA optimization.

specifies another relative gradient convergence criterion. For least squares problems and the TRUREG, LEVMAR, NRRIDG, and NEWRAP techniques, the criterion of Browne (1982) is used,

This criterion is not used by the other techniques. The default value is . The optional integer value determines the number of successive iterations for which the criterion must be satisfied before the process can be terminated.

specifies the relative parameter convergence criterion. Termination requires a small relative parameter change in subsequent iterations,

The default value is . The optional integer value determines the number of successive iterations for which the criterion must be satisfied before the process can be terminated.

specifies the parameter of the relative function and relative gradient termination criteria. The default value is . See the XCONV= option for details.

specifies that termination requires the maximum absolute gradient element of the Lagrange function to be small,

The default value is =1E5. The optional integer value determines the number of successive iterations for which the criterion must be satisfied before the process can be terminated.

specifies that termination requires the predicted objective function reduction to be small:

The default value is =1E6. This is the criterion used by the programs VMCWD and VF02AD (Powell 1982b). The optional integer value determines the number of successive iterations for which the criterion must be satisfied before the process can be terminated.

specifies that termination requires the normalized predicted objective function reduction to be small:

where is defined by the FSIZE= option. The default value is =1E8. The optional integer value determines the number of successive iterations for which the criterion must be satisfied before the process can be terminated.

specifies the scalar factor for the covariance matrix of parameter estimates. The scalar replaces the default value . For more details, see the section Approximate Standard Errors.

specifies that the Hessian matrix of the objective function (rather than the Hessian matrix of the Lagrange function) be used for computing the approximate covariance matrix of parameter estimates and, therefore, the approximate standard errors.

It is theoretically not correct to use the NOHLF option. However, since most implementations use the Hessian matrix of the objective function and not the Hessian matrix of the Lagrange function for computing approximate standard errors, the NOHLF option can be used to compare the results.

displays information about the starting values and final values of the optimization process.

displays the approximate Hessian matrix. If general linear or nonlinear constraints are active at the solution, the projected approximate Hessian matrix is also displayed.

displays the Hessian matrix. If general linear or nonlinear constraints are active at the solution, the projected Hessian matrix is also displayed.

displays the optimization history. The PHISTORY option is set automatically if the PALL or PRINT option is set.

displays the initial values and derivatives (if available). The PINIT option is set automatically if the PALL option is set.

displays the Jacobian matrix of nonlinear constraints specified by the NLINCON statement. The PNLCJAC option is set automatically if the PALL option is set.

displays the results of the optimization process, such as parameter estimates and constraints.

specifies that the initial step-size value for each line search (used by the QUANEW, CONGRA, or NEWRAP technique) cannot be larger than times the step-size value used in the former iteration. If the factor is not specified, the default value is . The DAMPSTEP option can prevent the line-search algorithm from repeatedly stepping into regions where some objective functions are difficult to compute or where they can lead to floating-point overflows during the computation of objective functions and their derivatives. The DAMPSTEP<=> option can prevent time-costly function calls during line searches with very small step sizes of objective functions. For more information about setting the start values of each line search, see the section Restricting the Step Length.

specifies the scaling version of the Hessian or crossproduct Jacobian matrix used in NRRIDG, TRUREG, LEVMAR, NEWRAP, or DBLDOG optimization. If HS is not equal to zero, the first iteration and each restart iteration set the diagonal scaling matrix :

where are the diagonal elements of the Hessian or crossproduct Jacobian matrix. In every other iteration, the diagonal scaling matrix is updated depending on the HS option:

HS=0

specifies that no scaling is done.

HS=1

specifies the Moré (1978) scaling update:

HS=2

specifies the Dennis, Gay, and Welsch (1981) scaling update:

HS=3

specifies that is reset in each iteration:

In the preceding equations, is the relative machine precision. The default is HS=1 for LEVMAR minimization and HS=0 otherwise. Scaling of the Hessian or crossproduct Jacobian can be time-consuming in the case where general linear constraints are active.

specifies a threshold for the Lagrange multiplier that decides whether an active inequality constraint remains active or can be deactivated. For maximization, must be greater than zero; for minimization, must be smaller than zero. The default is

where "" stands for maximization, "" stands for minimization, is the value of the absolute gradient criterion, and is the maximum absolute element of the (projected) gradient or .

specifies the range , , for active and violated boundary and linear constraints. If the point satisfies the condition

the constraint is recognized as an active constraint. Otherwise, the constraint is either an inactive inequality or a violated inequality or equality constraint. The default value is =1E8. During the optimization process, the introduction of rounding errors can force PROC NLP to increase the value of by factors of 10. If this happens, it is indicated by a message displayed in the log.

specifies a criterion , , used in the update of the QR decomposition that decides whether an active constraint is linearly dependent on a set of other active constraints. The default is =1E8. The larger becomes, the more the active constraints are recognized as being linearly dependent.

suppresses the computation and displayed output of the determinant and the inertia of the Hessian, crossproduct Jacobian, and covariance matrices. The inertia of a symmetric matrix are the numbers of negative, positive, and zero eigenvalues. For large applications, the NOEIGNUM option can save computer time.

specifies that the QUANEW or CONGRA algorithm is restarted with a steepest descent/ascent search direction after at most iterations, . Default values are as follows:

• CONGRA: UPDATE=PB: restart is done automatically so specification of is not used.

• CONGRA: UPDATEPB: , where is the number of parameters.

• QUANEW: is the largest integer available.

specifies the version of the quasi-Newton optimization technique with nonlinear constraints.

VS=1

specifies the update of the vector as in Powell (1978a, 1978b) (update like VF02AD).

VS=2

specifies the update of the vector as in Powell (1982a, 1982b) (update like VMCWD).

The default is VS=2.