Shared Concepts and Topics

specifies an absolute function convergence criterion: for minimization, termination requires . The default value of r is the negative square root of the largest double-precision value, which serves only as a protection against overflows.

specifies an absolute function difference convergence criterion:

The default value is r = 0. The optional integer value n specifies the number of successive iterations for which the criterion must be satisfied before the process can be terminated.

specifies an absolute gradient convergence criterion:

The default value is r = 1E–5. The optional integer value n specifies the number of successive iterations for which the criterion must be satisfied before the process can be terminated.

specifies an absolute parameter convergence criterion:

The default is r = 1E–8 for the NMSIMP technique and r = 0 otherwise. The optional integer value n specifies the number of successive iterations for which the criterion must be satisfied before the process can terminate.

specifies an absolute singularity criterion for the computation of the inertia (number of positive, negative, and zero eigenvalues) of the Hessian and its projected forms. The default value is the square root of the smallest positive double-precision value.

specifies that the initial step length value for each line search (used by the QUANEW, CONGRA, or NEWRAP technique) cannot be larger than r times the step length value used in the former iteration. If the DAMPSTEP option is specified but r is not specified, the default is r = 2. 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 could lead to floating-point overflows during the computation of objective functions and their derivatives. The DAMPSTEP= option can save time-consuming function calls during the line searches of objective functions that result in very small steps.

specifies a relative function convergence criterion:

The default is r , where FDIGITS is by default and is the machine precision. Some procedures, such as the GLIMMIX procedure, enable you to change the value with the FDIGITS= option in the PROC statement. The optional integer value n specifies the number of successive iterations for which the criterion must be satisfied before the process can terminate.

specifies a second function convergence criterion:

The default value is r = 1E–6 for the NMSIMP technique and r= 0 otherwise. The optional integer value n specifies the number of successive iterations for which the criterion must be satisfied before the process can terminate.

specifies the FSIZE parameter of the relative function and relative gradient termination criteria. The default value is r = 0. For more details, see the FCONV= and GCONV= options.

specifies a relative gradient convergence criterion:

The default value is r = 1E–8. The optional integer value n specifies the number of successive iterations for which the criterion must be satisfied before the process can terminate.

specifies another relative gradient convergence criterion:

The default value is r = 0. The optional integer value n specifies the number of successive iterations for which the criterion must be satisfied before the process can terminate.

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 0, the first iteration and each restart iteration set the diagonal scaling matrix :

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

In each scaling update, is the relative machine precision. The default value is HS=0. Scaling of the Hessian can be time-consuming in the case where general linear constraints are active.

specifies how the initial estimate of the approximate Hessian is defined for the quasi-Newton techniques QUANEW and DBLDOG. There are two alternatives:

By default (if you do not specify the option INHESSIAN=r), the initial estimate of the approximate Hessian is set to the multiple of the identity matrix r, where the scalar r is computed from the magnitude of the initial gradient.

reduces the length of the first trial step during the line search of the first iterations. For highly nonlinear objective functions, such as the EXP function, the default initial radius of the trust-region algorithm TRUREG or DBLDOG or the default step length of the line-search algorithms can result in arithmetic overflows. If this occurs, you should specify decreasing values of 0 < r < 1 such as INSTEP=1E–1, INSTEP=1E–2, INSTEP=1E–4, and so on, until the iteration starts successfully.

specifies a threshold r for the Lagrange multiplier that determines whether an active inequality constraint remains active or can be deactivated. For maximization, r must be greater than zero; for minimization, r must be smaller than zero. An active inequality constraint can be deactivated only if its Lagrange multiplier is less than the threshold value. The default value is

where "+" is for maximization, "–" is for minimization, ABSGCONV is the value of the absolute gradient criterion, and is the maximum absolute element of the gradient or the projected gradient.

specifies the range r for active and violated boundary constraints, where r 0. If the point satisfies the following condition, the constraint i is recognized as an active constraint:

Otherwise, the constraint i is either an inactive inequality or a violated inequality or equality constraint. The default value is r = 1E–8. During the optimization process, the introduction of rounding errors can force the optimization to increase the value of r by a factor of for some k > 0. If this happens, it is indicated by a message displayed in the log.

specifies a criterion r, where r 0, that is used in the update of the QR decomposition and that determines whether an active constraint is linearly dependent on a set of other active constraints. The default value is r = 1E–8. The larger r becomes, the more the active constraints are recognized as being linearly dependent. If the value of r is larger than 0.1, it is reset to 0.1.

specifies the line-search method for the CONGRA, QUANEW, and NEWRAP optimization techniques. See Fletcher (1987) for an introduction to line-search techniques. The value of i can be as follows. The default is LIS=2.

specifies the degree of accuracy that should be obtained by the line-search algorithms LIS=2 and LIS=3. Usually an imprecise line search is inexpensive and successful. For more difficult optimization problems, a more precise and expensive line search might be necessary (Fletcher 1987). The LIS=2 line-search method (which is the default for the NEWRAP, QUANEW, and CONGRA techniques) and the LIS=3 line-search method approach exact line search for small LSPRECISION= values. If you have numerical problems, try to decrease the LSPRECISION= value to obtain a more precise line search. The default values are shown in Table 34.

For more details, see Fletcher (1987).

specifies the maximum number i of function calls in the optimization process. The default values are as follows:

Optimization can terminate 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.

specifies the maximum number i of iterations in the optimization process. The default values are as follows:

These default values are also valid when i is specified as a missing value.

specifies an upper bound for the step length of the line-search algorithms during the first n iterations. By default, r is the largest double-precision value and n is the largest integer available. Setting this option can improve the speed of convergence for the CONGRA, QUANEW, and NEWRAP techniques.

specifies an upper limit of r seconds of CPU time for the optimization process. The time specified by the MAXTIME= option is checked only once at the end of each iteration. Therefore, the actual running time can be much longer than that specified by the MAXTIME= option. The actual running time includes the rest of the time needed to finish the iteration and the time needed to generate the output of the results. By default, CPU time is not limited.

specifies the minimum number of iterations. The default value is 0. If you request more iterations than are actually needed for convergence to a stationary point, the optimization algorithms can behave strangely. For example, the effect of rounding errors can prevent the algorithm from continuing for the required number of iterations.

specifies a relative singularity criterion r, where r > 0, for the computation of the inertia (number of positive, negative, and zero eigenvalues) of the Hessian and its projected forms. The default value is 1E–12.

suppresses output that is related to optimization, such as the iteration history. This option, along with all NLOPTIONS statement options for displayed output, are ignored by the GLIMMIX and HPMIXED procedures.

displays all optional output for optimization. This option is supported only by the CALIS and SURVEYPHREG procedures.

displays the optimization history. The PHISTORY option is implied if the PALL option is specified. The PHISTORY option is supported only by the CALIS and SURVEYPHREG procedures.

specifies that the QUANEW or CONGRA technique is restarted with a steepest search direction after at most i iterations, where i > 0. Default values are as follows:

specifies the singularity criterion r, 0 r 1, that is used for the inversion of the Hessian matrix. The default value is 1E–8.

specifies the fileref that contains the information needed for remote monitoring.

specifies the optimization technique. You can find additional information about choosing an optimization technique in the section Choosing an Optimization Algorithm. Valid values for the TECHNIQUE= option are as follows:

specifies the update method for the quasi-Newton, double-dogleg, or conjugate-gradient optimization technique. Not every update method can be used with each optimizer.

The following are the valid methods for the UPDATE= option:

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

The default is VERSION=2.

specifies a relative singularity criterion r, where r > 0, for the computation of the inertia (number of positive, negative, and zero eigenvalues) of the Hessian and its projected forms. The default value is r = 1E–8.

specifies the relative parameter convergence criterion:

The default value is r = 1E–8 for the NMSIMP technique and r = 0 otherwise. The optional integer value n specifies the number of successive iterations for which the criterion must be satisfied before the process can be terminated.

specifies the XSIZE parameter r of the relative parameter termination criterion, where r 0. The default value is r = 0. For more details, see the XCONV= option.