E04DGF/E04DGA : NAG Library, Mark 22

Note: before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

Note: this routine uses optional parameters to define choices in the problem specification and in the details of the algorithm. If you wish to use default settings for all of the optional parameters, you need only read Sections 1 to 9 of this document. If, however, you wish to reset some or all of the settings please refer to Section 10 for a detailed description of the algorithm and to Section 11 for a detailed description of the specification of the optional parameters.

E04DGA is a version of E04DGF that has additional parameters in order to make it safe for use in multithreaded applications (see Section 5). The initialization routine E04WBF must have been called before calling E04DGA.

Before calling E04DGA, or either of the option setting routines E04DJA or E04DKA, routine E04WBF must be called. The specification for E04WBF is:

E04WBF should be called with RNAME='E04DGA'. LCWSAV, LLWSAV, LIWSAV and LRWSAV, the declared lengths of CWSAV, LWSAV, IWSAV and RWSAV respectively, must satisfy:

LCWSAV≥1
LLWSAV≥120
LIWSAV≥610
LRWSAV≥475

The contents of the arrays CWSAV, LWSAV, IWSAV and RWSAV must not be altered between calling routines E04DGA, E04DJA, E04DKA and E04WBF.

For maximum reliability, it is preferable to provide all first partial derivatives. If all of the derivatives cannot be provided, you are recommended to obtain approximate values (using finite differences) by calling E04XAF/E04XAA from within OBJFUN. This is illustrated in Section 9 in E04DJF/E04DJA.

The method used by E04DGF/E04DGA is described in Section 10.

Gill P E and Murray W (1979) Conjugate-gradient methods for large-scale nonlinear optimization Technical Report SOL 79-15 Department of Operations Research, Stanford University

Gill P E, Murray W and Wright M H (1981) Practical Optimization Academic Press

If on entry IFAIL=0 or -1, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Note: E04DGF/E04DGA may return useful information for one or more of the following detected errors or warnings.

Errors or warnings detected by the routine:

To evaluate an ‘acceptable’ set of finite difference intervals using E04XAF/E04XAA requires 2 function evaluations per variable for a well-scaled problem and up to 6 function evaluations per variable for a badly scaled problem.

The document for E04DJF/E04DJA includes an example program to solve the same problem using some of the optional parameters described in Section 11.

Note: the remainder of this document is intended for more advanced users. Section 10 contains a detailed description of the algorithm which may be needed in order to understand Section 11. Section 11 describes the optional parameters which may be set by calls to E04DJF/E04DJA and/or E04DKF/E04DKA.

E04DGF/E04DGA uses a pre-conditioned conjugate gradient method and is based upon algorithm PLMA as described in Section 4.8.3 of Gill and Murray (1979) and Gill et al. (1981).

The algorithm proceeds as follows:

Let x0 be a given starting point and let k denote the current iteration, starting with k=0. The iteration requires gk, the gradient vector evaluated at xk, the kth estimate of the minimum. At each iteration a vector pk (known as the direction of search) is computed and the new estimate xk+1 is given by xk+αkpk where αk (the step length) minimizes the function Fxk+αkpk with respect to the scalar αk. A choice of initial step α0 is taken as

α0 = min1, 2 × Fk-Fest / gkT gk

where Fest is a user-supplied estimate of the function value at the solution. If Fest is not specified, the software always chooses the unit step length for α0. Subsequent step length estimates are computed using cubic interpolation with safeguards.

A quasi-Newton method can be used to compute the search direction pk by updating the inverse of the approximate Hessian Hk and computing

pk+1=-Hk+1gk+1.

(1)

The updating formula for the approximate inverse is given by

Hk+1 = Hk - 1 ykT sk Hk yk skT + sk ykT Hk + 1 ykT sk 1 + ykT Hk yk ykT sk sk skT ,

(2)

where yk=gk-1-gk and sk=xk+1-xk=αkpk.

The method used to obtain the search direction is based upon computing pk+1 as -Hk+1gk+1 where Hk+1 is a matrix obtained by updating the identity matrix with a limited number of quasi-Newton corrections. The storage of an n by n matrix is avoided by storing only the vectors that define the rank two corrections – hence the term ‘limited-memory’ quasi-Newton method. The precise method depends upon the number of updating vectors stored. For example, the direction obtained with the ‘one-step’ limited memory update is given by (1) using (2) with Hk equal to the identity matrix, viz.

pk+1 = - gk+1 + 1 ykT sk skT gk+1 yk + ykT gk+1 sk - skT gk+1 ykT sk 1 + ykT yk ykT sk sk .

Using a limited-memory quasi-Newton formula, such as the one above, guarantees pk+1 to be a descent direction if all the inner products ykT sk are positive for all vectors yk and sk used in the updating formula.

The remainder of this section can be skipped if you wish to use the default values for all optional parameters. A complete list of optional parameters and their default values is given in Section 11.1.

Optional parameters may be specified by calling one, or both, of the routines E04DJF/E04DJA and E04DKF/E04DKA before a call to E04DGF/E04DGA.

E04DJF/E04DJA reads options from an external options file, with Begin and End as the first and last lines respectively and each intermediate line defining a single optional parameter. For example,

Begin
   Print Level = 1
End

The call

 CALL E04DJF (IOPTNS, INFORM)

can then be used to read the file on unit IOPTNS. INFORM will be zero on successful exit. E04DJF/E04DJA should be consulted for a full description of this method of supplying optional parameters.

E04DKF/E04DKA can be called to supply options directly, one call being necessary for each optional parameter. For example,

 CALL E04DKF ('Print Level = 1')

E04DKF/E04DKA should be consulted for a full description of this method of supplying optional parameters.

All optional parameters not specified by you are set to their default values. Optional parameters specified by you are unaltered by E04DGF/E04DGA (unless they define invalid values) and so remain in effect for subsequent calls unless altered by you.

The following list gives the valid options. For each option, we give the keyword, any essential optional qualifiers and the default value. A definition for each option can be found in Section 11.2, where the minimum abbreviation of each keyword is underlined (if no characters of an optional qualifier are underlined, the qualifier may be omitted), the letter a denotes a phrase (character string) that qualifies an option, the letters i and r denote INTEGER and double precision values required with certain options, the number ε is a generic notation for machine precision (see X02AJF), and εr denotes the relative precision of the objective function.

This special keyword may be used to reset all optional parameters to their default values.

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 Section 10). 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 min1X02AMF,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 Section 5), 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 E04DGF/E04DGA, as indicated below. A detailed description of the printout is given in Section 8.1 (summary output at each iteration and the final solution).

i	Output
00	No output.
01	The final solution only.
05	One line of summary output (<80 characters; see Section 8.1) for each iteration (no printout of the final solution).
10	The final solution and one line of summary output for each iteration.

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:

i	Meaning
-1	No checks are performed.
-0	Only a ‘cheap’ test will be performed, requiring one call to OBJFUN.
-1	In addition to the ‘cheap’ test, individual gradient elements will also be checked using a reliable (but more expensive) test.

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

SUBROUTINE E04DGF (	N, OBJFUN, ITER, OBJF, OBJGRD, X, IWORK, WORK, IUSER, RUSER, IFAIL)
INTEGER	N, ITER, IWORK(N+1), IUSER(*), IFAIL
*double precision*	OBJF, OBJGRD(N), X(N), WORK(13N), RUSER()
EXTERNAL	OBJFUN

SUBROUTINE E04DGA (	N, OBJFUN, ITER, OBJF, OBJGRD, X, IWORK, WORK, IUSER, RUSER, LWSAV, IWSAV, RWSAV, IFAIL)
INTEGER	N, ITER, IWORK(N+1), IUSER(*), IWSAV(610), IFAIL
*double precision*	OBJF, OBJGRD(N), X(N), WORK(13N), RUSER(), RWSAV(475)
LOGICAL	LWSAV(120)
EXTERNAL	OBJFUN

SUBROUTINE E04WBF (	RNAME, CWSAV, LCWSAV, LWSAV, LLWSAV, IWSAV, LIWSAV, RWSAV, LRWSAV, IFAIL)
INTEGER	LCWSAV, LLWSAV, IWSAV(LIWSAV), LIWSAV, LRWSAV, IFAIL
*double precision*	RWSAV(LRWSAV)
LOGICAL	LWSAV(LLWSAV)
CHARACTER*6	RNAME
CHARACTER*80	CWSAV(LCWSAV)

SUBROUTINE OBJFUN (	MODE, N, X, OBJF, OBJGRD, NSTATE, IUSER, RUSER)
INTEGER	MODE, N, NSTATE, IUSER(*)
*double precision*	X(N), OBJF, OBJGRD(N), RUSER(*)

(i)	Fk-1-Fk<τF1+Fk
(ii)	xk-1-xk<τF1+xk
(iii)	gk≤τF31+Fk or gk<εA

NAG Library Routine Document

E04DGF/E04DGA

+− Contents

1 Purpose

2 Specification

2.1 Specification for E04DGF

2.2 Specification for E04DGA

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

8.1 Description of Printed Output

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

10 Algorithmic Details

11 Optional Parameters

11.1 Optional Parameter Checklist and Default Values

11.2 Description of the Optional Parameters

Itn	is the iteration count.
Step	is the step αk taken along the computed search direction. On reasonably well-behaved problems, the unit step (i.e., αk=1) will be taken as the solution is approached.
Nfun	is the cumulated number of evaluations of the objective function needed for the linesearch. Evaluations needed for the verification of the gradients by finite differences are not included. Nfun is printed as a guide to the amount of work required for the linesearch. E04DGF/E04DGA will perform at most 11 function evaluations per iteration.
Objective	is the value of the objective function at xk.
Norm G	is the Euclidean norm of the gradient of the objective function at xk.
Norm X	is the Euclidean norm of xk.
Norm (X(k-1)-X(k))	is the Euclidean norm of xk-1-xk.

Variable	gives the name (Varbl) and index j, for j=1,2,…,n of the variable.
Value	is the value of the variable at the final iteration.
Gradient Value	is the value of the gradient of the objective function with respect to the jth variable at the final iteration.

Optional Parameter	Default Value
Defaults
Estimated Optimal Function Value
Function Precision	Default =ε0.9
Iteration Limit	Default =max50,5n
Iters
Itns
Linesearch Tolerance	Default =0.9
List	Default for E04DGF=List
Maximum Step Length	Default =1020
Nolist	Default for E04DGA=Nolist
Optimality Tolerance	Default =εR0.8
Print Level	Default for E04DGF =10 Default for E04DGA =0
Start Objective Check at Variable	Default =1
Stop Objective Check at Variable	Default =n
Verify
Verify Gradients
Verify Level	Default =0
Verify Objective Gradients

NAG Library Routine DocumentE04DGF/E04DGA

+− Contents

NAG Library Routine Document

E04DGF/E04DGA