e05uc is the AD Library version of the primal routine e05ucf. Based (in the C++ interface) on overload resolution, e05uc can be used for primal, tangent and adjoint evaluation. It supports tangents and adjoints of first order.

2 Specification

Fortran Interface

Subroutine e05uc_AD_f (

n, nclin, ncnln, a, lda, bl, bu, confun, objfun, npts, x, ldx, start, repeat, nb, objf, objgrd, ldobjd, iter, c, ldc, cjac, ldcjac, sdcjac, r, ldr, sdr, clamda, ldclda, istate, listat, iopts, opts, iuser, ruser, info, ifail)

Integer, Intent (In)	::	n, nclin, ncnln, lda, npts, ldx, nb, ldobjd, ldc, ldcjac, sdcjac, ldr, sdr, ldclda, listat
Integer, Intent (Inout)	::	iopts(740), iuser(*), ifail
Integer, Intent (Out)	::	iter(nb), istate(listat,nb), info(nb)
ADTYPE, Intent (In)	::	a(lda,*), bl(n+nclin+ncnln), bu(n+nclin+ncnln)
ADTYPE, Intent (Inout)	::	x(ldx,nb), objgrd(ldobjd,nb), c(ldc,nb), cjac(ldcjac,sdcjac,nb), r(ldr,sdr,nb), clamda(ldclda,nb), opts(485), ruser(*)
ADTYPE, Intent (Out)	::	objf(nb)
Logical, Intent (In)	::	repeat
Type (c_ptr), Intent (Inout)	::	ad_handle
External	::	confun, objfun, start

Corresponding to the overloaded C++ function, the Fortran interface provides five routines with names reflecting the type used for active real arguments. The actual subroutine and type names are formed by replacing AD and ADTYPE in the above as follows:

when ADTYPE is	Real(kind=nag_wp)	then AD is p0w
when ADTYPE is	Type(nagad_a1w_w_rtype)	then AD is a1w
when ADTYPE is	Type(nagad_t1w_w_rtype)	then AD is t1w

C++ Interface

#include <dco.hpp>
#include <nagad.h>
namespace nag {
namespace ad {
template <typename CONFUN_T, typename OBJFUN_T, typename START_T>

void e05uc (

handle_t &ad_handle, const Integer &n, const Integer &nclin, const Integer &ncnln, const ADTYPE a[], const Integer &lda, const ADTYPE bl[], const ADTYPE bu[], CONFUN_T &&confun, OBJFUN_T &&objfun, const Integer &npts, ADTYPE x[], const Integer &ldx, START_T &&start, const logical &repeat, const Integer &nb, ADTYPE objf[], ADTYPE objgrd[], const Integer &ldobjd, Integer iter[], ADTYPE c[], const Integer &ldc, ADTYPE cjac[], const Integer &ldcjac, const Integer &sdcjac, ADTYPE r[], const Integer &ldr, const Integer &sdr, ADTYPE clamda[], const Integer &ldclda, Integer istate[], const Integer &listat, Integer iopts[], ADTYPE opts[], Integer info[], Integer &ifail)

}
}

The function is overloaded on ADTYPE which represents the type of active arguments. ADTYPE may be any of the following types:
double,
dco::ga1s<double>::type,
dco::gt1s<double>::type

Note: this function can be used with AD tools other than dco/c++. For details, please contact NAG.

3 Description

e05uc is the AD Library version of the primal routine e05ucf.

e05ucf is designed to find the global minimum of an arbitrary smooth function subject to constraints (which may include simple bounds on the variables, linear constraints and smooth nonlinear constraints) by generating a number of different starting points and performing a local search from each using sequential quadratic programming. For further information see Section 3 in the documentation for e05ucf.

4 References

Dennis J E Jr and Moré J J (1977) Quasi-Newton methods, motivation and theory SIAM Rev. 19 46–89

Dennis J E Jr and Schnabel R B (1981) A new derivation of symmetric positive-definite secant updates nonlinear programming (eds O L Mangasarian, R R Meyer and S M Robinson) 4 167–199 Academic Press

Dennis J E Jr and Schnabel R B (1983) Numerical Methods for Unconstrained Optimization and Nonlinear Equations Prentice–Hall

Fletcher R (1987) Practical Methods of Optimization (2nd Edition) Wiley

Gill P E, Hammarling S, Murray W, Saunders M A and Wright M H (1986) Users' guide for LSSOL (Version 1.0) Report SOL 86-1 Department of Operations Research, Stanford University

Gill P E, Murray W, Saunders M A and Wright M H (1984) Users' guide for SOL/QPSOL version 3.2 Report SOL 84–5 Department of Operations Research, Stanford University

Gill P E, Murray W, Saunders M A and Wright M H (1986a) Some theoretical properties of an augmented Lagrangian merit function Report SOL 86–6R Department of Operations Research, Stanford University

Gill P E, Murray W, Saunders M A and Wright M H (1986b) Users' guide for NPSOL (Version 4.0): a Fortran package for nonlinear programming Report SOL 86-2 Department of Operations Research, Stanford University

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

Powell M J D (1974) Introduction to constrained optimization Numerical Methods for Constrained Optimization (eds P E Gill and W Murray) 1–28 Academic Press

Powell M J D (1983) Variable metric methods in constrained optimization Mathematical Programming: the State of the Art (eds A Bachem, M Grötschel and B Korte) 288–311 Springer–Verlag

5 Arguments

In addition to the arguments present in the interface of the primal routine, e05uc includes some arguments specific to AD.

A brief summary of the AD specific arguments is given below. For the remainder, links are provided to the corresponding argument from the primal routine. A tooltip popup for all arguments can be found by hovering over the argument name in Section 2 and in this section.

1: ad_handle – nag::ad::handle_t Input/Output

On entry: a configuration object that holds information on the differentiation strategy. Details on setting the AD strategy are described in AD handle object in the NAG AD Library Introduction.

2: n – Integer Input

3: nclin – Integer Input

4: ncnln – Integer Input

5: a(lda, $*$ ) – ADTYPE array Input

6: lda – Integer Input

7: bl( $n + nclin + ncnln$ ) – ADTYPE array Input

8: bu( $n + nclin + ncnln$ ) – ADTYPE array Input

9: confun – Callable Input

confun needs to be callable with the specification listed below. This can be a C++ lambda, a functor or a (static member) function pointer. If using a lambda, parameters can be captured safely by reference. No copies of the callable are made internally.

If a null pointer is used as the argument, then a NAG supplied routine will be used as the argument for this parameter (C++ only).

For the Fortran interface, the NAG supplied routine e04ud_AD_m may be used as the actual argument for this parameter.

The specification of confun is:

Fortran Interface

Subroutine confun (

mode, ncnln, n, ldcjsl, needc, x, c, cjsl, nstate, iuser, ruser)

Integer, Intent (In)	::	ncnln, n, ldcjsl, needc(ncnln), nstate
Integer, Intent (Inout)	::	mode, iuser(*)
ADTYPE, Intent (In)	::	x(n)
ADTYPE, Intent (Inout)	::	cjsl(ldcjsl,n), ruser(*)
ADTYPE, Intent (Out)	::	c(ncnln)
Type (c_ptr), Intent (Inout)	::	ad_handle

C++ Interface

auto confun = [&](

const handle_t &ad_handle, Integer &mode, const Integer &ncnln, const Integer &n, const Integer &ldcjsl, const Integer needc[], const ADTYPE x[], ADTYPE c[], ADTYPE cjsl[], const Integer &nstate)

1: ad_handle – nag::ad::handle_t Input/Output: On entry: a handle to the AD handle object.
2: mode – Integer Input/Output
3: ncnln – Integer Input
4: n – Integer Input
5: ldcjsl – Integer Input
6: needc – Integer array Input
7: x – ADTYPE array Input
8: c – ADTYPE array Output
9: cjsl – ADTYPE array Input/Output
10: nstate – Integer Input
*: iuser – Integer array User Workspace
*: ruser – ADTYPE array User Workspace

10: objfun – Callable Input

objfun needs to be callable with the specification listed below. This can be a C++ lambda, a functor or a (static member) function pointer. If using a lambda, parameters can be captured safely by reference. No copies of the callable are made internally.

The specification of objfun is:

Fortran Interface

Subroutine objfun (

mode, n, x, objf, objgrd, nstate, iuser, ruser)

Integer, Intent (In)	::	n, nstate
Integer, Intent (Inout)	::	mode, iuser(*)
ADTYPE, Intent (In)	::	x(n)
ADTYPE, Intent (Inout)	::	objgrd(n), ruser(*)
ADTYPE, Intent (Out)	::	objf
Type (c_ptr), Intent (Inout)	::	ad_handle

C++ Interface

auto objfun = [&](

const handle_t &ad_handle, Integer &mode, const Integer &n, const ADTYPE x[], ADTYPE &objf, ADTYPE objgrd[], const Integer &nstate)

1: ad_handle – nag::ad::handle_t Input/Output: On entry: a handle to the AD handle object.
2: mode – Integer Input/Output
3: n – Integer Input
4: x – ADTYPE array Input
5: objf – ADTYPE Output
6: objgrd – ADTYPE array Input/Output
7: nstate – Integer Input
*: iuser – Integer array User Workspace
*: ruser – ADTYPE array User Workspace

11: npts – Integer Input

12: x(ldx, nb) – ADTYPE array Output

13: ldx – Integer Input

14: start – Callable Input

start needs to be callable with the specification listed below. This can be a C++ lambda, a functor or a (static member) function pointer. If using a lambda, parameters can be captured safely by reference. No copies of the callable are made internally.

If a null pointer is used as the argument, then a NAG supplied routine will be used as the argument for this parameter (C++ only).

For the Fortran interface, the NAG supplied routine e05uc_AD_z may be used as the actual argument for this parameter.

The specification of start is:

Fortran Interface

Subroutine start (

npts, quas, n, repeat, bl, bu, iuser, ruser, mode)

Integer, Intent (In)	::	npts, n
Integer, Intent (Inout)	::	iuser(*), mode
ADTYPE, Intent (In)	::	bl(n), bu(n)
ADTYPE, Intent (Inout)	::	quas(n,npts), ruser(*)
Logical, Intent (In)	::	repeat
Type (c_ptr), Intent (Inout)	::	ad_handle

C++ Interface

auto start = [&](

const handle_t &ad_handle, const Integer &npts, ADTYPE quas[], const Integer &n, const logical &repeat, const ADTYPE bl[], const ADTYPE bu[], Integer &mode)

1: ad_handle – nag::ad::handle_t Input/Output: On entry: a handle to the AD handle object.
2: npts – Integer Input
3: quas – ADTYPE array Input/Output
4: n – Integer Input
5: repeat – logical Input
6: bl – ADTYPE array Input
7: bu – ADTYPE array Input
*: iuser – Integer array User Workspace
*: ruser – ADTYPE array User Workspace
8: mode – Integer Input/Output

15: repeat – logical Input

16: nb – Integer Input

17: objf(nb) – ADTYPE array Output

18: objgrd(ldobjd, nb) – ADTYPE array Output

19: ldobjd – Integer Input

20: iter(nb) – Integer array Output

21: c(ldc, nb) – ADTYPE array Output

22: ldc – Integer Input

23: cjac(ldcjac, sdcjac, nb) – ADTYPE array Output

24: ldcjac – Integer Input

25: sdcjac – Integer Input

26: r(ldr, sdr, nb) – ADTYPE array Output

27: ldr – Integer Input

28: sdr – Integer Input

29: clamda(ldclda, nb) – ADTYPE array Output

30: ldclda – Integer Input

31: istate(listat, nb) – Integer array Output

32: listat – Integer Input

33: iopts( $740$ ) – Integer array Communication Array

34: opts( $485$ ) – ADTYPE array Communication Array

Please consult Overwriting of Inputs in the NAG AD Library Introduction.

*: iuser( $*$ ) – Integer array User Workspace

*: ruser( $*$ ) – ADTYPE array User Workspace

Please consult Overwriting of Inputs in the NAG AD Library Introduction.

35: info(nb) – Integer array Output

36: ifail – Integer Input/Output

6 Error Indicators and Warnings

e05uc preserves all error codes from e05ucf and in addition can return:

$ifail = - 89$: An unexpected AD error has been triggered by this routine. Please contact NAG.
See Error Handling in the NAG AD Library Introduction for further information.

$ifail = - 199$: The routine was called using a strategy that has not yet been implemented.
See AD Strategies in the NAG AD Library Introduction for further information.

$ifail = - 444$: A C++ exception was thrown.
The error message will show the details of the C++ exception text.

$ifail = - 899$: Dynamic memory allocation failed for AD.
See Error Handling in the NAG AD Library Introduction for further information.

7 Accuracy

Not applicable.

8 Parallelism and Performance

e05uc is not threaded in any implementation.

9 Further Comments

None.

10 Example

The following examples are variants of the example for e05ucf, modified to demonstrate calling the NAG AD Library.

Description of the primal example.

This example finds the global minimum of the two-dimensional Schwefel function:

\underset{x \in R^{2}}{minimize} f = \sum_{j = 1}^{2} x_{j} \sin (\sqrt{| x_{j} |})

subject to the constraints:

\begin{matrix} −10000 < 3.0 x_{1} - 2.0 x_{2} < 10.0, \\ - 1.0 < x_{1}^{2} - x_{2}^{2} + 3.0 x_{1} x_{2} < 500000.0, \\ - 0.9 < \cos ({(x_{1} / 200)}^{2} + (x_{2} / 100)) < 0.9, \\ −500 \leq x_{1} \leq 500, \\ −500 \leq x_{2} \leq 500 . \end{matrix}

10.1 Adjoint modes

Language	Source File	Data	Results
Fortran	e05uc_a1w_fe.f90	e05uc_a1w_fe.d	e05uc_a1w_fe.r
C++	e05uc_a1w_hcppe.cpp	e05uc_a1w_hcppe.d	e05uc_a1w_hcppe.r

10.2 Tangent modes

Language	Source File	Data	Results
Fortran	e05uc_t1w_fe.f90	e05uc_t1w_fe.d	e05uc_t1w_fe.r
C++	e05uc_t1w_hcppe.cpp	e05uc_t1w_hcppe.d	e05uc_t1w_hcppe.r

10.3 Passive mode

Language	Source File	Data	Results
Fortran	e05uc_p0w_fe.f90	e05uc_p0w_fe.d	e05uc_p0w_fe.r
C++	e05uc_p0w_hcppe.cpp	e05uc_p0w_hcppe.d	e05uc_p0w_hcppe.r

NAG Library Manual, Mark 28.4

Interfaces: FL CL CPP AD

NAG AD Library Introduction

E05 (Glopt) Chapter Contents

E05 (Glopt) Chapter Introduction (FL)

e05uc: FL CL CPP AD

NAG AD Librarye05uc (nlp_multistart_sqp)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Adjoint modes

10.2 Tangent modes

10.3 Passive mode

NAG AD Library
e05uc (nlp_multistart_sqp)