NAG AD Library
e04uc (nlp1_solve)

e04uc is the AD Library version of the primal routine e04ucf. Based (in the C++ interface) on overload resolution, e04uc can be used for primal, tangent and adjoint evaluation. It supports tangents and adjoints of first order.

2 Specification

Fortran Interface

Subroutine e04uc_AD_f (

ad_handle, n, nclin, ncnln, lda, ldcj, ldr, a, bl, bu, confun, objfun, iter, istate, c, cjac, clamda, objf, objgrd, r, x, iwork, liwork, work, lwork, iuser, ruser, lwsav, iwsav, rwsav, ifail)

Integer, Intent (In)	::	n, nclin, ncnln, lda, ldcj, ldr, liwork, lwork
Integer, Intent (Inout)	::	istate(n+nclin+ncnln), iuser(*), iwsav(610), ifail
Integer, Intent (Out)	::	iter, iwork(liwork)
ADTYPE, Intent (In)	::	a(lda,*), bl(n+nclin+ncnln), bu(n+nclin+ncnln)
ADTYPE, Intent (Inout)	::	cjac(ldcj,), clamda(n+nclin+ncnln), r(ldr,n), x(n), ruser(), rwsav(475)
ADTYPE, Intent (Out)	::	c(max(1,ncnln)), objf, objgrd(n), work(lwork)
Logical, Intent (Inout)	::	lwsav(120)
Type (c_ptr), Intent (Inout)	::	ad_handle
External	::	confun, objfun

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>

void e04uc (

handle_t &ad_handle, const Integer &n, const Integer &nclin, const Integer &ncnln, const Integer &lda, const Integer &ldcj, const Integer &ldr, const ADTYPE a[], const ADTYPE bl[], const ADTYPE bu[], CONFUN_T &&confun, OBJFUN_T &&objfun, Integer &iter, Integer istate[], ADTYPE c[], ADTYPE cjac[], ADTYPE clamda[], ADTYPE &objf, ADTYPE objgrd[], ADTYPE r[], ADTYPE x[], Integer iwork[], const Integer &liwork, ADTYPE work[], const Integer &lwork, logical lwsav[], Integer iwsav[], ADTYPE rwsav[], 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

e04uc is the AD Library version of the primal routine e04ucf.

e04ucf is designed to minimize an arbitrary smooth function subject to constraints (which may include simple bounds on the variables, linear constraints and smooth nonlinear constraints) using a Sequential Quadratic Programming (SQP) method. As many first derivatives as possible should be supplied by you; any unspecified derivatives are approximated by finite differences. It is not intended for large sparse problems.

e04ucf may also be used for unconstrained, bound-constrained and linearly constrained optimization.

e04ucf uses forward communication for evaluating the objective function, the nonlinear constraint functions, and any of their derivatives. For further information see Section 3 in the documentation for e04ucf.

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 (1984a) Procedures for optimization problems with a mixture of bounds and general linear constraints ACM Trans. Math. Software 10 282–298

Gill P E, Murray W, Saunders M A and Wright M H (1984b) 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

Hock W and Schittkowski K (1981) Test Examples for Nonlinear Programming Codes. Lecture Notes in Economics and Mathematical Systems 187 Springer–Verlag

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, e04uc 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: lda – Integer Input

6: ldcj – Integer Input

7: ldr – Integer Input

8: a(lda, $*$ ) – ADTYPE array Input

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

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

11: 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 (

ad_handle, mode, ncnln, n, ldcj, needc, x, c, cjac, nstate, iuser, ruser)

Integer, Intent (In)	::	ncnln, n, ldcj, needc(ncnln), nstate
Integer, Intent (Inout)	::	mode, iuser(*)
ADTYPE, Intent (In)	::	x(n)
ADTYPE, Intent (Inout)	::	cjac(ldcj,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 &ldcj, const Integer needc[], const ADTYPE x[], ADTYPE c[], ADTYPE cjac[], 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: ldcj – Integer Input
6: needc – Integer array Input
7: x – ADTYPE array Input
8: c – ADTYPE array Output
9: cjac – ADTYPE array Input/Output
10: nstate – Integer Input
*: iuser – Integer array User Workspace
*: ruser – ADTYPE array User Workspace

12: 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 (

ad_handle, 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

13: iter – Integer Output

14: istate( $n + nclin + ncnln$ ) – Integer array Input/Output

15: c( $\max (1, ncnln)$ ) – ADTYPE array Output

16: cjac(ldcj, $*$ ) – ADTYPE array Input/Output

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

17: clamda( $n + nclin + ncnln$ ) – ADTYPE array Input/Output

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

18: objf – ADTYPE Output

19: objgrd(n) – ADTYPE array Output

20: r(ldr, n) – ADTYPE array Input/Output

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

21: x(n) – ADTYPE array Input/Output

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

22: iwork(liwork) – Integer array Workspace

23: liwork – Integer Input

24: work(lwork) – ADTYPE array Workspace

25: lwork – Integer Input

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

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

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

26: lwsav( $120$ ) – logical array Communication Array

The arrays lwsav, iwsav and rwsav must not be altered between calls to any of the routines routine, routine or routine.

27: iwsav( $610$ ) – Integer array Communication Array

The arrays lwsav, iwsav and rwsav must not be altered between calls to any of the routines routine, routine or routine.

28: rwsav( $475$ ) – ADTYPE array Communication Array

The arrays lwsav, iwsav and rwsav must not be altered between calls to any of the routines routine, routine or routine.

29: ifail – Integer Input/Output

6 Error Indicators and Warnings

e04uc preserves all error codes from e04ucf 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

e04uc is not threaded in any implementation.

9 Further Comments

None.

10 Example

The following examples are variants of the example for e04ucf, modified to demonstrate calling the NAG AD Library.

Description of the primal example.

This is based on Problem 71 in Hock and Schittkowski (1981) and involves the minimization of the nonlinear function

F (x) = x_{1} x_{4} (x_{1} + x_{2} + x_{3}) + x_{3}

subject to the bounds

\begin{matrix} 1 \leq x_{1} \leq 5 \\ 1 \leq x_{2} \leq 5 \\ 1 \leq x_{3} \leq 5 \\ 1 \leq x_{4} \leq 5 \end{matrix}

to the general linear constraint

x_{1} + x_{2} + x_{3} + x_{4} \leq 20,

and to the nonlinear constraints

\begin{matrix} x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2} \leq 40, \\ x_{1} x_{2} x_{3} x_{4} \geq 25 . \end{matrix}

The initial point, which is infeasible, is

x_{0} = {(1, 5, 5, 1)}^{T},

and

F (x_{0}) = 16

The optimal solution (to five figures) is

x^{*} = {(1.0, 4.7430, 3.8211, 1.3794)}^{T},

and

F (x^{*}) = 17.014

. One bound constraint and both nonlinear constraints are active at the solution.

The document for e04ud includes an example program to solve the same problem using some of the optional parameters described in Section 12.

10.1 Adjoint modes

Language	Source File	Data	Results
Fortran	e04uc_a1w_fe.f90	e04uc_a1w_fe.d	e04uc_a1w_fe.r
C++	e04uc_a1w_hcppe.cpp	e04uc_a1w_hcppe.d	e04uc_a1w_hcppe.r

10.2 Tangent modes

Language	Source File	Data	Results
Fortran	e04uc_t1w_fe.f90	e04uc_t1w_fe.d	e04uc_t1w_fe.r
C++	e04uc_t1w_hcppe.cpp	e04uc_t1w_hcppe.d	e04uc_t1w_hcppe.r

10.3 Passive mode

Language	Source File	Data	Results
Fortran	e04uc_p0w_fe.f90	e04uc_p0w_fe.d	e04uc_p0w_fe.r
C++	e04uc_p0w_hcppe.cpp	e04uc_p0w_hcppe.d	e04uc_p0w_hcppe.r

NAG Library Manual, Mark 29.3

Interfaces: FL CL CPP AD

NAG AD Library Introduction

E04 (Opt) Chapter Contents

E04 (Opt) Chapter Introduction (FL)

e04uc: FL CL CPP AD

NAG AD Librarye04uc (nlp1_solve)

▸▿ 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
e04uc (nlp1_solve)