Nonlinear Optimization Methods 
The following table summarizes the options available in the NLO system.
Option 
Description 

Optimization Specifications 

TECHNIQUE= 
minimization technique 
UPDATE= 
update technique 
LINESEARCH= 
linesearch method 
LSPRECISION= 
linesearch precision 
HESCAL= 
type of Hessian scaling 
INHESSIAN= 
start for approximated Hessian 
RESTART= 
iteration number for update restart 
Termination Criteria Specifications 

MAXFUNC= 
maximum number of function calls 
MAXITER= 
maximum number of iterations 
MINITER= 
minimum number of iterations 
MAXTIME= 
upper limit seconds of CPU time 
ABSCONV= 
absolute function convergence criterion 
ABSFCONV= 
absolute function convergence criterion 
ABSGCONV= 
absolute gradient convergence criterion 
ABSXCONV= 
absolute parameter convergence criterion 
FCONV= 
relative function convergence criterion 
FCONV2= 
relative function convergence criterion 
GCONV= 
relative gradient convergence criterion 
XCONV= 
relative parameter convergence criterion 
FSIZE= 
used in FCONV, GCONV criterion 
XSIZE= 
used in XCONV criterion 
Step Length Options 

DAMPSTEP= 
damped steps in line search 
MAXSTEP= 
maximum trust region radius 
INSTEP= 
initial trust region radius 
Printed Output Options 

PALL 
display (almost) all printed optimizationrelated output 
PHISTORY 
display optimization history 
PHISTPARMS 
display parameter estimates in each iteration 
PSHORT 
reduce some default optimizationrelated output 
PSUMMARY 
reduce most default optimizationrelated output 
NOPRINT 
suppress all printed optimizationrelated output 
Remote Monitoring Options 

SOCKET= 
specify the fileref for remote monitoring 
These options are described in alphabetical order.
specifies an absolute function convergence criterion. For minimization, termination requires . The default value of is the negative square root of the largest doubleprecision value, which serves only as a protection against overflows.
specifies an absolute function convergence criterion. The default value is . The optional integer value 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 . The optional integer value 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 for the NMSIMP technique and otherwise. The optional integer value specifies the number of successive iterations for which the criterion must be satisfied before the process can terminate.
specifies that the initial step length value for each line search (used by the QUANEW, HYQUAN, CONGRA, or NEWRAP technique) cannot be larger than times the step length value used in the former iteration. If the DAMPSTEP option is specified but is not specified, the default is . The DAMPSTEP= option can prevent the linesearch 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 timecostly function calls during the line searches of objective functions that result in very small steps.
specifies a relative function convergence criterion. For all techniques except NMSIMP, termination requires a small relative change of the function value in successive iterations,
where FSIZE is defined by the FSIZE= option. The same formula is used for the NMSIMP technique, but is defined as the vertex with the lowest function value, and is defined as the vertex with the highest function value in the simplex. The default value may depend on the procedure. In most cases, you can use the PALL option to find it.
specifies another function convergence criterion.
For all techniques except NMSIMP, 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
For the NMSIMP technique, termination requires a small standard deviation of the function values of the simplex vertices , , where . If there are boundary constraints active at , the mean and standard deviation are computed only for the unconstrained vertices.
The default value is for the NMSIMP technique and otherwise. The optional integer value 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 . For more details, see the FCONV= and GCONV= options.
specifies a relative gradient convergence criterion. The default value is . The optional integer value 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 matrix used in NRRIDG, TRUREG, NEWRAP, or DBLDOG optimization.
If HS is not equal to 0, the first iteration and each restart iteration sets the diagonal scaling matrix :
where are the diagonal elements of the Hessian. In every other iteration, the diagonal scaling matrix is updated depending on the HS option:
specifies that no scaling is done.
specifies the Moré (1978) scaling update:
specifies the Dennis, Gay, & Welsch (1981) scaling update:
specifies that is reset in each iteration:
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 quasiNewton techniques QUANEW and DBLDOG. There are two alternatives:
If you do not use the specification, the initial estimate of the approximate Hessian is set to the Hessian at .
If you do use the specification, the initial estimate of the approximate Hessian is set to the multiple of the identity matrix .
By default, if you do not specify the option INHESSIAN=, the initial estimate of the approximate Hessian is set to the multiple of the identity matrix , where the scalar 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 trustregion algorithm TRUREG or DBLDOG or the default step length of the linesearch algorithms can result in arithmetic overflows. If this occurs, you should specify decreasing values of such as INSTEP=, INSTEP=, INSTEP=, and so on, until the iteration starts successfully.
For trustregion algorithms (TRUREG, DBLDOG), the INSTEP= option specifies a factor for the initial radius of the trust region. The default initial trustregion radius is the length of the scaled gradient. This step corresponds to the default radius factor of .
For linesearch algorithms (NEWRAP, CONGRA, QUANEW), the INSTEP= option specifies an upper bound for the initial step length for the line search during the first five iterations. The default initial step length is .
For the NelderMead simplex algorithm, using TECH=NMSIMP, the INSTEP= option defines the size of the start simplex.
specifies the linesearch method for the CONGRA, QUANEW, and NEWRAP optimization techniques. Refer to Fletcher (1987) for an introduction to linesearch techniques. The value of can be . For CONGRA, QUANEW and NEWRAP, the default value is .
specifies a linesearch method that needs the same number of function and gradient calls for cubic interpolation and cubic extrapolation; this method is similar to one used by the Harwell subroutine library.
specifies a linesearch method that needs more function than gradient calls for quadratic and cubic interpolation and cubic extrapolation; this method is implemented as shown in Fletcher (1987) and can be modified to an exact line search by using the LSPRECISION= option.
specifies a linesearch method that needs the same number of function and gradient calls for cubic interpolation and cubic extrapolation; this method is implemented as shown in Fletcher (1987) and can be modified to an exact line search by using the LSPRECISION= option.
specifies a linesearch method that needs the same number of function and gradient calls for stepwise extrapolation and cubic interpolation.
specifies a linesearch method that is a modified version of LIS=4.
specifies golden section line search (Polak 1971), which uses only function values for linear approximation.
specifies bisection line search (Polak 1971), which uses only function values for linear approximation.
specifies the Armijo linesearch technique (Polak 1971), which uses only function values for linear approximation.
specifies the degree of accuracy that should be obtained by the linesearch 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 may be necessary (Fletcher 1987). The second linesearch method (which is the default for the NEWRAP, QUANEW, and CONGRA techniques) and the third linesearch method approach exact line search for small LSPRECISION= values. If you have numerical problems, you should try to decrease the LSPRECISION= value to obtain a more precise line search. The default values are shown in the following table.
TECH= 
UPDATE= 
LSP default 

QUANEW 
DBFGS, BFGS 
= 0.4 
QUANEW 
DDFP, DFP 
= 0.06 
CONGRA 
all 
= 0.1 
NEWRAP 
no update 
= 0.9 
For more details, refer to Fletcher (1987).
specifies the maximum number of function calls in the optimization process. The default values are
TRUREG, NRRIDG, NEWRAP: 125
QUANEW, DBLDOG: 500
CONGRA: 1000
NMSIMP: 3000
Note that the optimization can terminate only after completing a full iteration. Therefore, the number of function calls that is actually performed can exceed the number that is specified by the MAXFUNC= option.
specifies the maximum number of iterations in the optimization process. The default values are
TRUREG, NRRIDG, NEWRAP: 50
QUANEW, DBLDOG: 200
CONGRA: 400
NMSIMP: 1000
These default values are also valid when is specified as a missing value.
specifies an upper bound for the step length of the linesearch algorithms during the first iterations. By default, is the largest doubleprecision value and 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 seconds of CPU time for the optimization process. The default value is the largest floatingpoint double representation of your computer. Note that 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.
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.
suppresses the output. (See procedure documentation for availability of this option.)
displays all optional output for optimization. (See procedure documentation for availability of this option.)
displays the optimization history. (See procedure documentation for availability of this option.)
display parameter estimates in each iteration. (See procedure documentation for availability of this option.)
displays the initial values and derivatives (if available). (See procedure documentation for availability of this option.)
restricts the amount of default output. (See procedure documentation for availability of this option.)
restricts the amount of default displayed output to a short form of iteration history and notes, warnings, and errors. (See procedure documentation for availability of this option.)
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 performed automatically, is not used.
CONGRA
UPDATEPB: , where is the number of parameters.
QUANEW
is the largest integer available.
Specifies the fileref that contains the information needed for remote monitoring. See the section Remote Monitoring for more details.
specifies the optimization technique. Valid values are as follows:
CONGRA
performs a conjugategradient optimization, which can be more precisely specified with the UPDATE= option and modified with the LINESEARCH= option. When you specify this option, UPDATE=PB by default.
DBLDOG
performs a version of doubledogleg optimization, which can be more precisely specified with the UPDATE= option. When you specify this option, UPDATE=DBFGS by default.
NMSIMP
performs a NelderMead simplex optimization.
NONE
does not perform any optimization. This option can be used as follows:
to perform a grid search without optimization
to compute estimates and predictions that cannot be obtained efficiently with any of the optimization techniques
NEWRAP
performs a NewtonRaphson optimization that combines a linesearch algorithm with ridging. The linesearch algorithm LIS=2 is the default method.
NRRIDG
performs a NewtonRaphson optimization with ridging.
QUANEW
performs a quasiNewton optimization, which can be defined more precisely with the UPDATE= option and modified with the LINESEARCH= option. This is the default estimation method.
TRUREG
performs a trust region optimization.
specifies the update method for the QUANEW, DBLDOG, or CONGRA optimization technique. Not every update method can be used with each optimizer.
Valid methods are as follows:
BFGS
performs the original Broyden, Fletcher, Goldfarb, and Shanno (BFGS) update of the inverse Hessian matrix.
DBFGS
performs the dual BFGS update of the Cholesky factor of the Hessian matrix. This is the default update method.
DDFP
performs the dual Davidon, Fletcher, and Powell (DFP) update of the Cholesky factor of the Hessian matrix.
DFP
performs the original DFP update of the inverse Hessian matrix.
PB
performs the automatic restart update method of Powell (1977) and Beale (1972).
FR
performs the FletcherReeves update (Fletcher 1987).
PR
performs the PolakRibiere update (Fletcher 1987).
CD
performs a conjugatedescent update of Fletcher (1987).
specifies the relative parameter convergence criterion. For all techniques except NMSIMP, termination requires a small relative parameter change in subsequent iterations. The default value is for the NMSIMP technique and otherwise. The optional integer value specifies the number of successive iterations for which the criterion must be satisfied before the process can be terminated.
specifies the XSIZE parameter of the relative parameter termination criterion. The default value is . For more detail, see the XCONV= option.
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