This value of r specifies the user-supplied guess of the optimum objective function value Fest. This value is used to calculate an initial step length α0 (see [Algorithmic Details]). If the value of r is not specified (the default), then this has the effect of setting α0 to unity. It should be noted that for badly scaled functions a unit step along the steepest descent direction will often compute the objective function at very large values of x.

The parameter defines εr, which is intended to be a measure of the accuracy with which the problem function Fx can be computed. If r<ε or r≥1, the default value is used.

The value of εr should reflect the relative precision of 1+Fx; i.e., εr acts as a relative precision when F is large, and as an absolute precision when F is small. For example, if Fx is typically of order 1000 and the first six significant digits are known to be correct, an appropriate value for εr would be 10-6. In contrast, if Fx is typically of order 10-4 and the first six significant digits are known to be correct, an appropriate value for εr would be 10-10. The choice of εr can be quite complicated for badly scaled problems; see Chapter 8 of Gill et al. (1981) for a discussion of scaling techniques. The default value is appropriate for most simple functions that are computed with full accuracy. However when the accuracy of the computed function values is known to be significantly worse than full precision, the value of εr should be large enough so that no attempt will be made to distinguish between function values that differ by less than the error inherent in the calculation.

The value of i specifies the maximum number of iterations allowed before termination. If i<0, the default value is used.

Problems whose Hessian matrices at the solution contain sets of clustered eigenvalues are likely to be minimized in significantly fewer than n iterations. Problems without this property may require anything between n and 5n iterations, with approximately 2n iterations being a common figure for moderately difficult problems.

The value r controls the accuracy with which the step α taken during each iteration approximates a minimum of the function along the search direction (the smaller the value of r, the more accurate the linesearch). The default value r=0.9 requests an inaccurate search, and is appropriate for most problems. A more accurate search may be appropriate when it is desirable to reduce the number of iterations – for example, if the objective function is cheap to evaluate. If r<0 or r≥1, the default value is used.

Normally each optional parameter specification is printed as it is supplied. Optional parameter Nolist may be used to suppress the printing and optional parameter List may be used to restore printing.

If r>0, the maximum allowable step length for the linesearch is taken as min1x02am,rpk . If r≤0, the default value is used.

The parameter r specifies the accuracy to which you wish the final iterate to approximate a solution of the problem. Broadly speaking, r indicates the number of correct figures desired in the objective function at the solution. For example, if r is 10-6 and termination occurs with ifail=0 (see [Parameters]), then the final point satisfies the termination criteria, where τF represents Optimality Tolerance. If r<εr or r≥1, the default value is used.

The value i controls the amount of printout produced by e04dg, as indicated below. A detailed description of the printout is given in [Description of Printed Output] (summary output at each iteration and the final solution).

These keywords take effect only if Verify Level>0. They may be used to control the verification of gradient elements computed by objfun. For example, if the first 30 elements of the objective gradient appeared to be correct in an earlier run, so that only element 31 remains questionable, it is reasonable to specify Start Objective Check at Variable=31. If the first 30 variables appear linearly in the objective, so that the corresponding gradient elements are constant, the above choice would also be appropriate.

If i1≤0 or i1>max1,minn,i2, the default value is used. If i2≤0 or i2>n, the default value is used.

These keywords refer to finite-difference checks on the gradient elements computed by objfun. Gradients are verified at the user-supplied initial estimate of the solution. The possible choices for i are as follows:

For example, the objective gradient will be verified if Verify, Verify=Yes, Verify Gradients, Verify Objective Gradients or Verify Level=1 is specified.