The NLP Procedure |
All optimization techniques stop iterating at
if at least one of a set of termination
criteria is satisfied.
PROC NLP also terminates if the point
is fully
constrained by
linearly independent active linear or
boundary constraints, and all Lagrange multiplier estimates
of active inequality constraints are greater than a small
negative tolerance.
Since the Nelder-Mead simplex algorithm does not use
derivatives, no termination criterion is available based
on the gradient of the objective function.
Powell's COBYLA algorithm uses only one more
termination criterion. COBYLA is a trust region algorithm
that sequentially reduces the radius of a spherical
trust region from a start radius
= INSTEP to the final radius
= ABSXTOL. The default value is
e-4. The convergence to small values
of
(high precision) may take many calls of
the function and constraint modules and may result in
numerical problems.
In some applications, the small default value of the ABSGCONV= criterion is too difficult to satisfy for some of the optimization techniques. This occurs most often when finite-difference approximations of derivatives are used.
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.
The QUANEW algorithm for nonlinearly constrained optimization
does not monotonically
reduce the value of either the objective function or some
kind of merit function which combines objective and constraint
functions. Furthermore, the algorithm uses the watchdog
technique with backtracking (Chamberlain et al. 1982).
Therefore, no termination criteria were implemented that are
based on the values ( or
) of successive iterations.
In addition to the criteria used by all optimization
techniques, three more termination criteria are currently
available. They are based on satisfying the Karush-Kuhn-Tucker conditions,
which require that the gradient of the Lagrange function is zero at
the optimal point
:
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