NAG Library Routine Document

E04USF/E04USA

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, to Section 11 for a detailed description of the specification of the optional parameters and to Section 12 for a detailed description of the monitoring information produced by the routine.

+− Contents

1 Purpose

E04USF/E04USA is designed to minimize an arbitrary smooth sum of squares 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. See the description of the optional parameter Derivative Level, in Section 11.2. It is not intended for large sparse problems.

E04USF/E04USA may also be used for unconstrained, bound-constrained and linearly constrained optimization.

E04USA is a version of E04USF 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 E04USA.

2 Specification

2.1 Specification for E04USF

SUBROUTINE E04USF (	M, N, NCLIN, NCNLN, LDA, LDCJ, LDFJ, LDR, A, BL, BU, Y, CONFUN, OBJFUN, ITER, ISTATE, C, CJAC, F, FJAC, CLAMDA, OBJF, R, X, IWORK, LIWORK, WORK, LWORK, IUSER, RUSER, IFAIL)
INTEGER	M, N, NCLIN, NCNLN, LDA, LDCJ, LDFJ, LDR, ITER, ISTATE(N+NCLIN+NCNLN), IWORK(LIWORK), LIWORK, LWORK, IUSER(*), IFAIL
*double precision*	A(LDA,), BL(N+NCLIN+NCNLN), BU(N+NCLIN+NCNLN), Y(M), C(max(1,NCNLN)), CJAC(LDCJ,), F(M), FJAC(LDFJ,N), CLAMDA(N+NCLIN+NCNLN), OBJF, R(LDR,N), X(N), WORK(LWORK), RUSER(*)
EXTERNAL	CONFUN, OBJFUN

2.2 Specification for E04USA

SUBROUTINE E04USA (	M, N, NCLIN, NCNLN, LDA, LDCJ, LDFJ, LDR, A, BL, BU, Y, CONFUN, OBJFUN, ITER, ISTATE, C, CJAC, F, FJAC, CLAMDA, OBJF, R, X, IWORK, LIWORK, WORK, LWORK, IUSER, RUSER, LWSAV, IWSAV, RWSAV, IFAIL)
INTEGER	M, N, NCLIN, NCNLN, LDA, LDCJ, LDFJ, LDR, ITER, ISTATE(N+NCLIN+NCNLN), IWORK(LIWORK), LIWORK, LWORK, IUSER(*), IWSAV(610), IFAIL
*double precision*	A(LDA,), BL(N+NCLIN+NCNLN), BU(N+NCLIN+NCNLN), Y(M), C(max(1,NCNLN)), CJAC(LDCJ,), F(M), FJAC(LDFJ,N), CLAMDA(N+NCLIN+NCNLN), OBJF, R(LDR,N), X(N), WORK(LWORK), RUSER(*), RWSAV(475)
LOGICAL	LWSAV(120)
EXTERNAL	CONFUN, OBJFUN

Before calling E04USA, or either of the option setting routines E04UQA or E04URA, E04WBF must be called. The specification for E04WBF is:

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)

E04WBF should be called with RNAME='E04USA'. 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 E04UQA, E04URA, E04USA and E04WBF.

3 Description

E04USF/E04USA is designed to solve the nonlinear least-squares programming problem – the minimization of a smooth nonlinear sum of squares function subject to a set of constraints on the variables. The problem is assumed to be stated in the following form:

Minimizex∈RnFx=12∑i=1myi-fix2 subject to l≤ x ALx cx ≤u,

(1)

where Fx (the objective function) is a nonlinear function which can be represented as the sum of squares of m subfunctions y1-f1x,y2-f2x,…,ym-fmx, the yi are constant, AL is an nL by n constant matrix, and cx is an nN element vector of nonlinear constraint functions. (The matrix AL and the vector cx may be empty.) The objective function and the constraint functions are assumed to be smooth, i.e., at least twice-continuously differentiable. (The method of E04USF/E04USA will usually solve (1) if there are only isolated discontinuities away from the solution.)

Note that although the bounds on the variables could be included in the definition of the linear constraints, we prefer to distinguish between them for reasons of computational efficiency. For the same reason, the linear constraints should not be included in the definition of the nonlinear constraints. Upper and lower bounds are specified for all the variables and for all the constraints. An equality constraint can be specified by setting li=ui. If certain bounds are not present, the associated elements of l or u can be set to special values that will be treated as -∞ or +∞. (See the description of the optional parameter Infinite Bound Size.)

You must supply an initial estimate of the solution to (1), together with subroutines that define fx = f1x,f2x,…,fmxT , cx and as many first partial derivatives as possible; unspecified derivatives are approximated by finite differences.

The subfunctions are defined by the array Y and OBJFUN, and the nonlinear constraints are defined by CONFUN. On every call, these subroutines must return appropriate values of fx and cx. You should also provide the available partial derivatives. Any unspecified derivatives are approximated by finite differences for a discussion of the optional parameter Derivative Level. Note that if there are any nonlinear constraints, then the first call to CONFUN will precede the first call to OBJFUN.

For maximum reliability, it is preferable for you to provide all partial derivatives (see Chapter 8 of Gill et al. (1981) for a detailed discussion). If all gradients cannot be provided, it is similarly advisable to provide as many as possible. While developing OBJFUN and CONFUN, the optional parameter Verify should be used to check the calculation of any known gradients.

4 References

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

5 Parameters

1: M – INTEGERInput

On entry: m, the number of subfunctions associated with Fx.

Constraint: M>0.

2: N – INTEGERInput

On entry: n, the number of variables.

Constraint: N>0.

3: NCLIN – INTEGERInput

On entry: nL, the number of general linear constraints.

Constraint: NCLIN≥0.

4: NCNLN – INTEGERInput

On entry: nN, the number of nonlinear constraints.

Constraint: NCNLN≥0.

5: LDA – INTEGERInput

On entry: the first dimension of the array A as declared in the (sub)program from which E04USF/E04USA is called.

Constraint: LDA≥max1,NCLIN.

6: LDCJ – INTEGERInput

On entry: the first dimension of the array CJAC as declared in the (sub)program from which E04USF/E04USA is called.

Constraint: LDCJ≥max1,NCNLN.

7: LDFJ – INTEGERInput

On entry: the first dimension of the array FJAC as declared in the (sub)program from which E04USF/E04USA is called.

Constraint: LDFJ≥M.

8: LDR – INTEGERInput

On entry: the first dimension of the array R as declared in the (sub)program from which E04USF/E04USA is called.

Constraint: LDR≥N.

9: A(LDA,*) – double precision arrayInput

Note: the second dimension of the array A must be at least N if NCLIN>0, and at least 1 otherwise.

On entry: the ith row of A contains the ith row of the matrix AL of general linear constraints in (1). That is, the ith row contains the coefficients of the ith general linear constraint, for i=1,2,…,NCLIN.

If NCLIN=0, the array A is not referenced.

10: BL(N+NCLIN+NCNLN) – double precision arrayInput

11: BU(N+NCLIN+NCNLN) – double precision arrayInput

On entry: must contain the lower bounds and BU the upper bounds, for all the constraints in the following order. The first n elements of each array must contain the bounds on the variables, the next nL elements the bounds for the general linear constraints (if any) and the next nN elements the bounds for the general nonlinear constraints (if any). To specify a nonexistent lower bound (i.e., lj=-∞), set BLj≤-bigbnd, and to specify a nonexistent upper bound (i.e., uj=+∞), set BUj≥bigbnd; the default value of bigbnd is 1020, but this may be changed by the optional parameter Infinite Bound Size. To specify the jth constraint as an equality, set BLj=BUj=β, say, where β<bigbnd.

Constraints:

BLj≤BUj, for j=1,2,…,N+NCLIN+NCNLN;
if BLj=BUj=β, β<bigbnd.

12: Y(M) – double precision arrayInput

On entry: the coefficients of the constant vector y of the objective function.

13: CONFUN – SUBROUTINE, supplied by the NAG Library or the user.External Procedure

CONFUN must calculate the vector cx of nonlinear constraint functions and (optionally) its Jacobian (= ∂c ∂x ) for a specified n element vector x. If there are no nonlinear constraints (i.e., NCNLN=0), CONFUN will never be called by E04USF/E04USA and CONFUN may be the dummy routine E04UDM. (E04UDM is included in the NAG Library.) If there are nonlinear constraints, the first call to CONFUN will occur before the first call to OBJFUN.

The specification of CONFUN is:

SUBROUTINE CONFUN (	MODE, NCNLN, N, LDCJ, NEEDC, X, C, CJAC, NSTATE, IUSER, RUSER)
INTEGER	MODE, NCNLN, N, LDCJ, NEEDC(NCNLN), NSTATE, IUSER(*)
*double precision*	X(N), C(NCNLN), CJAC(LDCJ,N), RUSER(*)

1: MODE – INTEGERInput/Output

On entry: indicates which values must be assigned during each call of CONFUN. Only the following values need be assigned, for each value of i such that NEEDCi>0:

MODE=0: Ci.
MODE=1: All available elements in the ith row of CJAC.
MODE=2: Ci and all available elements in the ith row of CJAC.

On exit: may be set to a negative value if you wish to terminate the solution to the current problem, and in this case E04USF/E04USA will terminate with IFAIL set to MODE.

2: NCNLN – INTEGERInput

On entry: nN, the number of nonlinear constraints.

3: N – INTEGERInput

On entry: n, the number of variables.

4: LDCJ – INTEGERInput

On entry: the first dimension of the array CJAC as declared in the (sub)program from which E04USF/E04USA is called.

5: NEEDC(NCNLN) – INTEGER arrayInput

On entry: the indices of the elements of C and/or CJAC that must be evaluated by CONFUN. If NEEDCi>0, then the ith element of C and/or the available elements of the ith row of CJAC (see parameter MODE) must be evaluated at x.

6: X(N) – double precision arrayInput

On entry: x, the vector of variables at which the constraint functions and/or all available elements of the constraint Jacobian are to be evaluated.

7: C(NCNLN) – double precision arrayOutput

On exit: if NEEDCi>0 and MODE=0 or 2, Ci must contain the value of the ith constraint at x. The remaining elements of C, corresponding to the nonpositive elements of NEEDC, are ignored.

8: CJAC(LDCJ,N) – double precision arrayInput/Output

On entry: is set to a special value.

On exit: if NEEDCi>0 and MODE=1 or 2, the ith row of CJAC must contain the available elements of the vector ∇ci given by

∇ci= ∂ci ∂x1 , ∂ci ∂x2 ,…, ∂ci ∂xn T,

where ∂ci ∂xj is the partial derivative of the ith constraint with respect to the jth variable, evaluated at the point x. See also the parameter NSTATE. The remaining rows of CJAC, corresponding to nonpositive elements of NEEDC, are ignored.

If all elements of the constraint Jacobian are known (i.e., Derivative Level=2 or 3), any constant elements may be assigned to CJAC one time only at the start of the optimization. An element of CJAC that is not subsequently assigned in CONFUN will retain its initial value throughout. Constant elements may be loaded into CJAC either before the call to E04USF/E04USA or during the first call to CONFUN (signalled by the value NSTATE=1). The ability to preload constants is useful when many Jacobian elements are identically zero, in which case CJAC may be initialized to zero and nonzero elements may be reset by CONFUN.

Note that constant nonzero elements do affect the values of the constraints. Thus, if CJACij is set to a constant value, it need not be reset in subsequent calls to CONFUN, but the value CJACij×Xj must nonetheless be added to Ci. For example, if CJAC11=2 and CJAC12=-5, then the term 2×X1-5×X2 must be included in the definition of C1.

It must be emphasised that, if Derivative Level=0 or 1, unassigned elements of CJAC are not treated as constant; they are estimated by finite differences, at nontrivial expense. If you do not supply a value for the optional parameter Difference Interval, an interval for each element of x is computed automatically at the start of the optimization. The automatic procedure can usually identify constant elements of CJAC, which are then computed once only by finite differences.

9: NSTATE – INTEGERInput

On entry: if NSTATE=1 then E04USF/E04USA is calling CONFUN for the first time. This parameter setting allows you to save computation time if certain data must be read or calculated only once.

10: IUSER(*) – INTEGER arrayUser Workspace

11: RUSER(*) – double precision arrayUser Workspace

CONFUN is called from E04USF/E04USA with the parameters IUSER and RUSER as supplied to E04USF/E04USA. You are free to use the arrays IUSER and RUSER to supply information to CONFUN as an alternative to using COMMON global variables.

CONFUN must be declared as EXTERNAL in the (sub)program from which E04USF/E04USA is called. Parameters denoted as Input must not be changed by this procedure.

Note: CONFUN should be tested separately before being used in conjunction with E04USF/E04USA. See also the description of the optional parameter Verify.

14: OBJFUN – SUBROUTINE, supplied by the user.External Procedure

OBJFUN must calculate either the ith element of the vector fx = f1x,f2x,…,fmxT or all m elements of fx and (optionally) its Jacobian (= ∂f ∂x ) for a specified n element vector x.

The specification of OBJFUN is:

SUBROUTINE OBJFUN (	MODE, M, N, LDFJ, NEEDFI, X, F, FJAC, NSTATE, IUSER, RUSER)
INTEGER	MODE, M, N, LDFJ, NEEDFI, NSTATE, IUSER(*)
*double precision*	X(N), F(M), FJAC(LDFJ,N), RUSER(*)

1: MODE – INTEGERInput/Output

On entry: indicates which values must be assigned during each call of OBJFUN. Only the following values need be assigned:

MODE=0 and NEEDFI=i>0: Fi.
MODE=0 and NEEDFI<0: F.
MODE=1 and NEEDFI<0: All available elements of FJAC.
MODE=2 and NEEDFI<0: F and all available elements of FJAC.

On exit: may be set to a negative value if you wish to terminate the solution to the current problem, and in this case E04USF/E04USA will terminate with IFAIL set to MODE.

2: M – INTEGERInput

On entry: m, the number of subfunctions.

3: N – INTEGERInput

On entry: n, the number of variables.

4: LDFJ – INTEGERInput

On entry: the first dimension of the array FJAC as declared in the (sub)program from which E04USF/E04USA is called.

5: NEEDFI – INTEGERInput

On entry: if NEEDFI=i>0, only the ith element of fx needs to be evaluated at x; the remaining elements need not be set. This can result in significant computational savings when m≫n.

6: X(N) – double precision arrayInput

On entry: x, the vector of variables at which fx and/or all available elements of its Jacobian are to be evaluated.

7: F(M) – double precision arrayOutput

On exit: if MODE=0 and NEEDFI=i>0, Fi must contain the value of fi at x.

If MODE=0 or 2 and NEEDFI<0, Fi must contain the value of fi at x, for i=1,2,…,m.

8: FJAC(LDFJ,N) – double precision arrayInput/Output

On entry: is set to a special value.

On exit: if MODE=1 or 2 and NEEDFI<0, the ith row of FJAC must contain the available elements of the vector ∇fi given by

∇fi= ∂fi ∂x1 , ∂fi ∂x2 ,…, ∂fi ∂xn T,

evaluated at the point x. See also the parameter NSTATE.

9: NSTATE – INTEGERInput

On entry: if NSTATE=1 then E04USF/E04USA is calling OBJFUN for the first time. This parameter setting allows you to save computation time if certain data must be read or calculated only once.

10: IUSER(*) – INTEGER arrayUser Workspace

11: RUSER(*) – double precision arrayUser Workspace

OBJFUN is called from E04USF/E04USA with the parameters IUSER and RUSER as supplied to E04USF/E04USA. You are free to use the arrays IUSER and RUSER to supply information to OBJFUN as an alternative to using COMMON global variables.

OBJFUN must be declared as EXTERNAL in the (sub)program from which E04USF/E04USA is called. Parameters denoted as Input must not be changed by this procedure.

Note: OBJFUN should be tested separately before being used in conjunction with E04USF/E04USA. See also the description of the optional parameter Verify.

15: ITER – INTEGEROutput

On exit: the number of major iterations performed.

16: ISTATE(N+NCLIN+NCNLN) – INTEGER arrayInput/Output

On entry: need not be set if the (default) optional parameter Cold Start is used.

If the optional parameter Warm Start has been chosen, the elements of ISTATE corresponding to the bounds and linear constraints define the initial working set for the procedure that finds a feasible point for the linear constraints and bounds. The active set at the conclusion of this procedure and the elements of ISTATE corresponding to nonlinear constraints then define the initial working set for the first QP subproblem. More precisely, the first n elements of ISTATE refer to the upper and lower bounds on the variables, the next nL elements refer to the upper and lower bounds on ALx, and the next nN elements refer to the upper and lower bounds on cx. Possible values for ISTATEj are as follows:

ISTATEj	Meaning
0	The corresponding constraint is not in the initial QP working set.
1	This inequality constraint should be in the working set at its lower bound.
2	This inequality constraint should be in the working set at its upper bound.
3	This equality constraint should be in the initial working set. This value must not be specified unless BLj=BUj.

The values -2, -1 and 4 are also acceptable but will be modified by the routine. If E04USF/E04USA has been called previously with the same values of N, NCLIN and NCNLN, ISTATE already contains satisfactory information. (See also the description of the optional parameter Warm Start.) The routine also adjusts (if necessary) the values supplied in X to be consistent with ISTATE.

Constraint: -2≤ISTATEj≤4, for j=1,2,…,N+NCLIN+NCNLN.

On exit: the status of the constraints in the QP working set at the point returned in X. The significance of each possible value of ISTATEj is as follows:

ISTATEj	Meaning
-2	This constraint violates its lower bound by more than the appropriate feasibility tolerance (see the optional parameters Linear Feasibility Tolerance and Nonlinear Feasibility Tolerance). This value can occur only when no feasible point can be found for a QP subproblem.
-1	This constraint violates its upper bound by more than the appropriate feasibility tolerance (see the optional parameters Linear Feasibility Tolerance and Nonlinear Feasibility Tolerance). This value can occur only when no feasible point can be found for a QP subproblem.
-0	The constraint is satisfied to within the feasibility tolerance, but is not in the QP working set.
-1	This inequality constraint is included in the QP working set at its lower bound.
-2	This inequality constraint is included in the QP working set at its upper bound.
-3	This constraint is included in the QP working set as an equality. This value of ISTATE can occur only when BLj=BUj.

17: C(max1,NCNLN) – double precision arrayOutput

On exit: if NCNLN>0, Ci contains the value of the ith nonlinear constraint function ci at the final iterate, for i=1,2,…,NCNLN.

If NCNLN=0, the array C is not referenced.

18: CJAC(LDCJ,*) – double precision arrayInput/Output

Note: the second dimension of the array CJAC must be at least N if NCNLN>0, and at least 1 otherwise.

On entry: in general, CJAC need not be initialized before the call to E04USF/E04USA. However, if Derivative Level=3, you may optionally set the constant elements of CJAC (see parameter NSTATE in the description of CONFUN). Such constant elements need not be re-assigned on subsequent calls to CONFUN.

On exit: if NCNLN>0, CJAC contains the Jacobian matrix of the nonlinear constraint functions at the final iterate, i.e., CJACij contains the partial derivative of the ith constraint function with respect to the jth variable, for i=1,2,…,NCNLN and j=1,2,…,N . (See the discussion of parameter CJAC under CONFUN.)

If NCNLN=0, the array CJAC is not referenced.

19: F(M) – double precision arrayOutput

On exit: Fi contains the value of the ith function fi at the final iterate, for i=1,2,…,M.

20: FJAC(LDFJ,N) – double precision arrayInput/Output

On entry: in general, FJAC need not be initialized before the call to E04USF/E04USA. However, if Derivative Level=3, you may optionally set the constant elements of FJAC (see parameter NSTATE in the description of OBJFUN). Such constant elements need not be re-assigned on subsequent calls to OBJFUN.

On exit: the Jacobian matrix of the functions f1,f2,…,fm at the final iterate, i.e., FJACij contains the partial derivative of the ith function with respect to the jth variable, for i=1,2…,M and j=1,2,…,N . (See also the discussion of parameter FJAC under OBJFUN.)

21: CLAMDA(N+NCLIN+NCNLN) – double precision arrayInput/Output

On entry: need not be set if the (default) optional parameter Cold Start is used.

If the optional parameter Warm Start has been chosen, CLAMDAj must contain a multiplier estimate for each nonlinear constraint with a sign that matches the status of the constraint specified by the ISTATE array, for j=N+NCLIN+1,N+NCLIN+2,…, N+NCLIN+NCNLN . The remaining elements need not be set. Note that if the jth constraint is defined as ‘inactive’ by the initial value of the ISTATE array (i.e., ISTATEj=0), CLAMDAj should be zero; if the jth constraint is an inequality active at its lower bound (i.e., ISTATEj=1), CLAMDAj should be non-negative; if the jth constraint is an inequality active at its upper bound (i.e., ISTATEj=2, CLAMDAj should be nonpositive. If necessary, the routine will modify CLAMDA to match these rules.

On exit: the values of the QP multipliers from the last QP subproblem. CLAMDAj should be non-negative if ISTATEj=1 and nonpositive if ISTATEj=2.

22: OBJF – double precisionOutput

On exit: the value of the objective function at the final iterate.

23: R(LDR,N) – double precision arrayInput/Output

On entry: need not be initialized if the (default) optional parameter Cold Start is used.

If the optional parameter Warm Start has been chosen, R must contain the upper triangular Cholesky factor R of the initial approximation of the Hessian of the Lagrangian function, with the variables in the natural order. Elements not in the upper triangular part of R are assumed to be zero and need not be assigned.

On exit: if Hessian=No, R contains the upper triangular Cholesky factor R of QTH~Q, an estimate of the transformed and reordered Hessian of the Lagrangian at x (see (6) in E04UFF/E04UFA). If Hessian=Yes, R contains the upper triangular Cholesky factor R of H, the approximate (untransformed) Hessian of the Lagrangian, with the variables in the natural order.

24: X(N) – double precision arrayInput/Output

On entry: an initial estimate of the solution.

On exit: the final estimate of the solution.

25: IWORK(LIWORK) – INTEGER arrayWorkspace

26: LIWORK – INTEGERInput

On entry: the dimension of the array IWORK as declared in the (sub)program from which E04USF/E04USA is called.

Constraint: LIWORK≥3×N+NCLIN+2×NCNLN.

27: WORK(LWORK) – double precision arrayWorkspace

28: LWORK – INTEGERInput

On entry: the dimension of the array WORK as declared in the (sub)program from which E04USF/E04USA is called.

Constraints:

if NCNLN=0 and NCLIN=0, LWORK≥20×N+M×N+3;
if NCNLN=0 and NCLIN>0, LWORK≥2×N2+20×N+11×NCLIN+ M×N+3;
if NCNLN>0 and NCLIN≥0, LWORK≥2×N2+N×NCLIN+2×N× NCNLN+20×N+11×NCLIN+21×NCNLN+M×N+3.

The amounts of workspace provided and required are (by default) output on the current advisory message unit (as defined by X04ABF). As an alternative to computing LIWORK and LWORK from the formulas given above, you may prefer to obtain appropriate values from the output of a preliminary run with LIWORK and LWORK set to 1. (E04USF/E04USA will then terminate with IFAIL=9.)

29: IUSER(*) – INTEGER arrayUser Workspace

Note: the dimension of the array IUSER must be at least 1.

IUSER is not used by E04USF/E04USA, but is passed directly to user-supplied subroutines CONFUN and OBJFUN and may be used to pass information to those routines.

30: RUSER(*) – double precision arrayUser Workspace

Note: the dimension of the array RUSER must be at least 1.

RUSER is not used by E04USF/E04USA, but is passed directly to user-supplied subroutines CONFUN and OBJFUN and may be used to pass information to those routines.

31: IFAIL – INTEGERInput/Output

Note: for E04USA, IFAIL does not occur in this position in the parameter list. See the additional parameters described below.

On entry: IFAIL must be set to 0, -1 or 1. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

On exit: IFAIL=0 unless the routine detects an error (see Section 6).

For environments where it might be inappropriate to halt program execution when an error is detected, the value -1 or 1 is recommended. If the output of error messages is undesirable, then the value 1 is recommended. Otherwise, because for this routine the values of the output parameters may be useful even if IFAIL≠0 on exit, the recommended value is -1. When the value -1 or 1 is used it is essential to test the value of IFAIL on exit.

E04USF/E04USA returns with IFAIL=0 if the iterates have converged to a point x that satisfies the first-order Kuhn–Tucker conditions (see Section 10.1 in E04UFF/E04UFA) to the accuracy requested by the optional parameter Optimality Tolerance (default value=εr0.8, where εr is the value of the optional parameter Function Precision (default value=ε0.9, where ε is the machine precision)), i.e., the projected gradient and active constraint residuals are negligible at x.

You should check whether the following four conditions are satisfied:

(i)	the final value of Norm Gz (see Section 8.1) is significantly less than that at the starting point;
(ii)	during the final major iterations, the values of Step and Mnr (see Section 8.1) are both one;
(iii)	the last few values of both Norm Gz and Violtn (see Section 8.1) become small at a fast linear rate; and
(iv)	Cond Hz (see Section 8.1) is small.

If all these conditions hold, x is almost certainly a local minimum of (1).

Note: the following are additional parameters for specific use with E04USA. Users of E04USF therefore need not read the remainder of this description.

31: LWSAV(120) – LOGICAL arrayCommunication Array

32: IWSAV(610) – INTEGER arrayCommunication Array

33: RWSAV(475) – double precision arrayCommunication Array

The arrays LWSAV, IWSAV and RWSAV must not be altered between calls to any of the routines E04WBF, E04USA, E04UQA or E04URA.

34: IFAIL – INTEGERInput/Output

Note: see the parameter description for IFAIL above.

6 Error Indicators and Warnings

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

Note: E04USF/E04USA may return useful information for one or more of the following detected errors or warnings.

Errors or warnings detected by the routine:

IFAIL<0: A negative value of IFAIL indicates an exit from E04USF/E04USA because you set MODE<0 in OBJFUN or CONFUN. The value of IFAIL will be the same as your setting of MODE.

IFAIL=1: The final iterate x satisfies the first-order Kuhn–Tucker conditions (see Section 10.1 in E04UFF/E04UFA) to the accuracy requested, but the sequence of iterates has not yet converged. E04USF/E04USA was terminated because no further improvement could be made in the merit function (see Section 8.1).

This value of IFAIL may occur in several circumstances. The most common situation is that you ask for a solution with accuracy that is not attainable with the given precision of the problem (as specified by the optional parameter Function Precision (default value=ε0.9, where ε is the machine precision)). This condition will also occur if, by chance, an iterate is an ‘exact’ Kuhn–Tucker point, but the change in the variables was significant at the previous iteration. (This situation often happens when minimizing very simple functions, such as quadratics.)

If the four conditions listed in Section 5 for IFAIL=0 are satisfied, x is likely to be a solution of (1) even if IFAIL=1.

IFAIL=2: E04USF/E04USA has terminated without finding a feasible point for the linear constraints and bounds, which means that either no feasible point exists for the given value of the optional parameter Linear Feasibility Tolerance (default value=ε, where ε is the machine precision), or no feasible point could be found in the number of iterations specified by the optional parameter Minor Iteration Limit (default value=max50,3n+nL+nN). You should check that there are no constraint redundancies. If the data for the constraints are accurate only to an absolute precision σ, you should ensure that the value of the optional parameter Linear Feasibility Tolerance is greater than σ. For example, if all elements of AL are of order unity and are accurate to only three decimal places, Linear Feasibility Tolerance should be at least 10-3.

IFAIL=3: No feasible point could be found for the nonlinear constraints. The problem may have no feasible solution. This means that there has been a sequence of QP subproblems for which no feasible point could be found (indicated by I at the end of each line of intermediate printout produced by the major iterations; see Section 8.1). This behaviour will occur if there is no feasible point for the nonlinear constraints. (However, there is no general test that can determine whether a feasible point exists for a set of nonlinear constraints.) If the infeasible subproblems occur from the very first major iteration, it is highly likely that no feasible point exists. If infeasibilities occur when earlier subproblems have been feasible, small constraint inconsistencies may be present. You should check the validity of constraints with negative values of ISTATE. If you are convinced that a feasible point does exist, E04USF/E04USA should be restarted at a different starting point.

IFAIL=4: The limiting number of iterations (as determined by the optional parameter Major Iteration Limit (default value=max50,3n+nL+10nN) has been reached.

If the algorithm appears to be making satisfactory progress, then Major Iteration Limit may be too small. If so, either increase its value and rerun E04USF/E04USA or, alternatively, rerun E04USF/E04USA using the optional parameter Warm Start. If the algorithm seems to be making little or no progress however, then you should check for incorrect gradients or ill-conditioning as described under IFAIL=6.

Note that ill-conditioning in the working set is sometimes resolved automatically by the algorithm, in which case performing additional iterations may be helpful. However, ill-conditioning in the Hessian approximation tends to persist once it has begun, so that allowing additional iterations without altering R is usually inadvisable. If the quasi-Newton update of the Hessian approximation was reset during the latter major iterations (i.e., an R occurs at the end of each line of intermediate printout; see Section 8.1), it may be worthwhile to try a Warm Start at the final point as suggested above.

IFAIL=5: Not used by this routine.

IFAIL=6: x does not satisfy the first-order Kuhn–Tucker conditions (see Section 10.1 in E04UFF/E04UFA), and no improved point for the merit function (see Section 8.1) could be found during the final linesearch.

This sometimes occurs because an overly stringent accuracy has been requested, i.e., the value of the optional parameter Optimality Tolerance (default value=εr0.8, where εr is the value of the optional parameter Function Precision (default value=ε0.9, where ε is the machine precision)) is too small. In this case you should apply the four tests described under IFAIL=0 to determine whether or not the final solution is acceptable (see Gill et al. (1981), for a discussion of the attainable accuracy).

If many iterations have occurred in which essentially no progress has been made and E04USF/E04USA has failed completely to move from the initial point then user-supplied subroutines OBJFUN and/or CONFUN may be incorrect. You should refer to comments under IFAIL=7 and check the gradients using the optional parameter Verify (default value=0). Unfortunately, there may be small errors in the objective and constraint gradients that cannot be detected by the verification process. Finite difference approximations to first derivatives are catastrophically affected by even small inaccuracies. An indication of this situation is a dramatic alteration in the iterates if the finite difference interval is altered. One might also suspect this type of error if a switch is made to central differences even when Norm Gz and Violtn (see Section 8.1) are large.
Another possibility is that the search direction has become inaccurate because of ill-conditioning in the Hessian approximation or the matrix of constraints in the working set; either form of ill-conditioning tends to be reflected in large values of Mnr (the number of iterations required to solve each QP subproblem; see Section 8.1).

If the condition estimate of the projected Hessian (Cond Hz; see Section 12) is extremely large, it may be worthwhile rerunning E04USF/E04USA from the final point with the optional parameter Warm Start. In this situation, ISTATE and CLAMDA should be left unaltered and R should be reset to the identity matrix.

If the matrix of constraints in the working set is ill-conditioned (i.e., Cond T is extremely large; see Section 12), it may be helpful to run E04USF/E04USA with a relaxed value of the optional parameter Feasibility Tolerance (default value=ε, where ε is the machine precision). (Constraint dependencies are often indicated by wide variations in size in the diagonal elements of the matrix T, whose diagonals will be printed if Major Print Level≥30).

IFAIL=7: The user-supplied derivatives of the subfunctions and/or nonlinear constraints appear to be incorrect.

Large errors were found in the derivatives of the subfunctions and/or nonlinear constraints. This value of IFAIL will occur if the verification process indicated that at least one Jacobian element had no correct figures. You should refer to the printed output to determine which elements are suspected to be in error.

As a first-step, you should check that the code for the subfunction and constraint values is correct – for example, by computing the subfunctions at a point where the correct value of Fx is known. However, care should be taken that the chosen point fully tests the evaluation of the subfunctions. It is remarkable how often the values x=0 or x=1 are used to test function evaluation procedures, and how often the special properties of these numbers make the test meaningless.

Special care should be used in this test if computation of the subfunctions involves subsidiary data communicated in COMMON storage. Although the first evaluation of the subfunctions may be correct, subsequent calculations may be in error because some of the subsidiary data has accidentally been overwritten.

Gradient checking will be ineffective if the objective function uses information computed by the constraints, since they are not necessarily computed before each function evaluation.

Errors in programming the subfunctions may be quite subtle in that the subfunction values are ‘almost’ correct. For example, a subfunction may not be accurate to full precision because of the inaccurate calculation of a subsidiary quantity, or the limited accuracy of data upon which the subfunction depends. A common error on machines where numerical calculations are usually performed in double precision is to include even one single precision constant in the calculation of the subfunction; since some compilers do not convert such constants to double precision, half the correct figures may be lost by such a seemingly trivial error.

IFAIL=8: Not used by this routine.

IFAIL=9: An input parameter is invalid.

overflow: If the printed output before the overflow error contains a warning about serious ill-conditioning in the working set when adding the jth constraint, it may be possible to avoid the difficulty by increasing the magnitude of the optional parameter Linear Feasibility Tolerance and/or the optional parameter Nonlinear Feasibility Tolerance and rerunning the program. If the message recurs even after this change then the offending linearly dependent constraint (with index ‘j’) must be removed from the problem. If overflow occurs in one of the user-supplied subroutines (e.g., if the nonlinear functions involve exponentials or singularities), it may help to specify tighter bounds for some of the variables (i.e., reduce the gap between the appropriate lj and uj).

7 Accuracy

If IFAIL=0 on exit, then the vector returned in the array X is an estimate of the solution to an accuracy of approximately Optimality Tolerance (default value=ε0.8, where ε is the machine precision).

8 Further Comments

8.1 Description of the Printed Output

This section describes the intermediate printout and final printout produced by E04USF/E04USA. The intermediate printout is a subset of the monitoring information produced by the routine at every iteration (see Section 12). You can control the level of printed output (see the description of the optional parameter Major Print Level). Note that the intermediate printout and final printout are produced only if Major Print Level≥10 (the default for E04USF, by default no output is produced by E04USA).

The following line of summary output (<80 characters) is produced at every major iteration. In all cases, the values of the quantities printed are those in effect on completion of the given iteration.

Maj	is the major iteration count.
Mnr	is the number of minor iterations required by the feasibility and optimality phases of the QP subproblem. Generally, Mnr will be 1 in the later iterations, since theoretical analysis predicts that the correct active set will be identified near the solution (see Section 10 in E04UFF/E04UFA). Note that Mnr may be greater than the optional parameter Minor Iteration Limit if some iterations are required for the feasibility phase.
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.
Merit Function	is the value of the augmented Lagrangian merit function (see (12) in E04UFF/E04UFA) at the current iterate. This function will decrease at each iteration unless it was necessary to increase the penalty parameters (see Section 10.3 in E04UFF/E04UFA). As the solution is approached, Merit Function will converge to the value of the objective function at the solution. If the QP subproblem does not have a feasible point (signified by I at the end of the current output line) then the merit function is a large multiple of the constraint violations, weighted by the penalty parameters. During a sequence of major iterations with infeasible subproblems, the sequence of Merit Function values will decrease monotonically until either a feasible subproblem is obtained or E04USF/E04USA terminates with IFAIL=3 (no feasible point could be found for the nonlinear constraints). If there are no nonlinear constraints present (i.e., NCNLN=0) then this entry contains Objective, the value of the objective function Fx. The objective function will decrease monotonically to its optimal value when there are no nonlinear constraints.
Norm Gz	is ZTgFR, the Euclidean norm of the projected gradient (see Section 10.2 in E04UFF/E04UFA). Norm Gz will be approximately zero in the neighbourhood of a solution.
Violtn	is the Euclidean norm of the residuals of constraints that are violated or in the predicted active set (not printed if NCNLN is zero). Violtn will be approximately zero in the neighbourhood of a solution.
Cond Hz	is a lower bound on the condition number of the projected Hessian approximation HZ ( HZ = ZT HFR Z = RZT RZ ; see (6) and (11) in E04UFF/E04UFA). The larger this number, the more difficult the problem.
M	is printed if the quasi-Newton update has been modified to ensure that the Hessian approximation is positive-definite (see Section 10.4 in E04UFF/E04UFA).
I	is printed if the QP subproblem has no feasible point.
C	is printed if central differences have been used to compute the unspecified objective and constraint gradients. If the value of Step is zero then the switch to central differences was made because no lower point could be found in the linesearch. (In this case, the QP subproblem is resolved with the central difference gradient and Jacobian.) If the value of Step is nonzero then central differences were computed because Norm Gz and Violtn imply that x is close to a Kuhn–Tucker point (see Section 10.1 in E04UFF/E04UFA).
L	is printed if the linesearch has produced a relative change in x greater than the value defined by the optional parameter Step Limit. If this output occurs frequently during later iterations of the run, optional parameter Step Limit should be set to a larger value.
R	is printed if the approximate Hessian has been refactorized. If the diagonal condition estimator of R indicates that the approximate Hessian is badly conditioned then the approximate Hessian is refactorized using column interchanges. If necessary, R is modified so that its diagonal condition estimator is bounded.

The final printout includes a listing of the status of every variable and constraint.

The following describes the printout for each variable. A full stop (.) is printed for any numerical value that is zero.

Varbl

gives the name (V) and index j, for j=1,2,…,n, of the variable.

State

gives the state of the variable (FR if neither bound is in the working set, EQ if a fixed variable, LL if on its lower bound, UL if on its upper bound, TF if temporarily fixed at its current value). If Value lies outside the upper or lower bounds by more than the Feasibility Tolerance, State will be ++ or -- respectively.

A key is sometimes printed before State.

A	Alternative optimum possible. The variable is active at one of its bounds, but its Lagrange multiplier is essentially zero. This means that if the variable were allowed to start moving away from its bound then there would be no change to the objective function. The values of the other free variables might change, giving a genuine alternative solution. However, if there are any degenerate variables (labelled D), the actual change might prove to be zero, since one of them could encounter a bound immediately. In either case the values of the Lagrange multipliers might also change.
D	Degenerate. The variable is free, but it is equal to (or very close to) one of its bounds.
I	Infeasible. The variable is currently violating one of its bounds by more than the Feasibility Tolerance.

Value

is the value of the variable at the final iteration.

Lower Bound

is the lower bound specified for the variable. None indicates that BLj≤-bigbnd.

Upper Bound

is the upper bound specified for the variable. None indicates that BUj≥bigbnd.

Lagr Mult

is the Lagrange multiplier for the associated bound. This will be zero if State is FR unless BLj≤-bigbnd and BUj≥bigbnd, in which case the entry will be blank. If x is optimal, the multiplier should be non-negative if State is LL and nonpositive if State is UL.

Slack

is the difference between the variable Value and the nearer of its (finite) bounds BLj and BUj. A blank entry indicates that the associated variable is not bounded (i.e., BLj≤-bigbnd and BUj≥bigbnd).

The meaning of the printout for linear and nonlinear constraints is the same as that given above for variables, with ‘variable’ replaced by ‘constraint’, BLj and BUj are replaced by BLn+j and BUn+j respectively, and with the following changes in the heading:

L Con	gives the name (L) and index j, for j=1,2,…,nL, of the linear constraint.
N Con	gives the name (N) and index (j-nL), for j=nL+1,nL+2,…,nL+nN, of the nonlinear constraint.

Note that movement off a constraint (as opposed to a variable moving away from its bound) can be interpreted as allowing the entry in the Slack column to become positive.

Numerical values are output with a fixed number of digits; they are not guaranteed to be accurate to this precision.

9 Example

This example is based on Problem 57 in Hock and Schittkowski (1981) and involves the minimization of the sum of squares function

Fx = 12 ∑ i=1 44 yi - fi x 2 ,

where

fi x = x1 + 0.49-x1 e -x2 ai- 8

and

i yi ai i yi ai 1 0.49 8 23 0.41 22 2 0.49 8 24 0.40 22 3 0.48 10 25 0.42 24 4 0.47 10 26 0.40 24 5 0.48 10 27 0.40 24 6 0.47 10 28 0.41 26 7 0.46 12 29 0.40 26 8 0.46 12 30 0.41 26 9 0.45 12 31 0.41 28 10 0.43 12 32 0.40 28 11 0.45 14 33 0.40 30 12 0.43 14 34 0.40 30 13 0.43 14 35 0.38 30 14 0.44 16 36 0.41 32 15 0.43 16 37 0.40 32 16 0.43 16 38 0.40 34 17 0.46 18 39 0.41 36 18 0.45 18 40 0.38 36 19 0.42 20 41 0.40 38 20 0.42 20 42 0.40 38 21 0.43 20 43 0.39 40 22 0.41 22 44 0.39 42

subject to the bounds

x1≥-0.4 x2≥-4.0

to the general linear constraint

x1+x2≥1.0

and to the nonlinear constraint

0.49x2-x1 x2≥ 0.09 .

The initial point, which is infeasible, is

x0=0.4,0.0T

and Fx0=0.002241.

The optimal solution (to five figures) is

x*=0.41995,1.28484T,

and Fx*=0.01423. The nonlinear constraint is active at the solution.

The document for E04UQF/E04UQA 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 11 describes the optional parameters which may be set by calls to E04UQF/E04UQA and/or E04URF/E04URA. Section 12 describes the quantities which can be requested to monitor the course of the computation.

10 Algorithmic Details

E04USF/E04USA implements a sequential quadratic programming (SQP) method incorporating an augmented Lagrangian merit function and a BFGS (Broyden–Fletcher–Goldfarb–Shanno) quasi-Newton approximation to the Hessian of the Lagrangian, and is based on E04WDF. The documents for E04NCF/E04NCA, E04UFF/E04UFA and E04WDF should be consulted for details of the method.

11 Optional Parameters

Several optional parameters in E04USF/E04USA define choices in the problem specification or the algorithm logic. In order to reduce the number of formal parameters of E04USF/E04USA these optional parameters have associated default values that are appropriate for most problems. Therefore you need only specify those optional parameters whose values are to be different from their default values.

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 E04UQF/E04UQA and E04URF/E04URA before a call to E04USF/E04USA.

E04UQF/E04UQA 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 E04UQF (IOPTNS, INFORM)

can then be used to read the file on unit IOPTNS. INFORM will be zero on successful exit. E04UQF/E04UQA should be consulted for a full description of this method of supplying optional parameters.

E04URF/E04URA can be called to supply options directly, one call being necessary for each optional parameter. For example,

 CALL E04URF ('Print Level = 1')

E04URF/E04URA 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 E04USF/E04USA (unless they define invalid values) and so remain in effect for subsequent calls to E04USF/E04USA, unless altered by you.

11.1 Optional Parameter Checklist and Default Values

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 (the Function Precision). Further details of other quantities not explicitly defined in this section may be found by consulting the document for E04UFF/E04UFA.

Optional Parameter	Default Value
Central Difference Interval	Default values are computed
Cold Start	Default
Crash Tolerance	Default =0.01
Defaults
Derivative Level	Default =3
Difference Interval	Default values are computed
Feasibility Tolerance	Default =ε
Function Precision	Default =ε0.9
Hessian	Default =No
Infinite Bound Size	Default =1020
Infinite Step Size	Default =maxbigbnd,1020
Iteration Limit
Iters
Itns
JTJ Initial Hessian	Default
Linear Feasibility Tolerance	Default =ε
Line Search Tolerance	Default =0.9
List	Default for E04USF=List
Major Iteration Limit	Default =max50,3n+nL+10nN
Major Print Level	Default for E04USF =10
Minor Iteration Limit	Default =max50,3n+nL+nN
Minor Print Level	Default =0
Monitoring File	Default =-1
Nolist	Default for E04USA=Nolist
Nonlinear Feasibility Tolerance	Default =ε0.33 or ε
Optimality Tolerance	Default =εR0.8
Print Level	Default for E04USA =0
Reset Frequency	Default =2
Start Constraint Check At Variable	Default =1
Start Objective Check At Variable	Default =1
Step Limit	Default =2.0
Stop Constraint Check At Variable	Default =n
Stop Objective Check At Variable	Default =n
Unit Initial Hessian
Verify
Verify Constraint Gradients
Verify Gradients
Verify Level	Default =0
Verify Objective Gradients
Warm Start

11.2 Description of the Optional Parameters

Central Difference Interval

Default values are computed

If the algorithm switches to central differences because the forward-difference approximation is not sufficiently accurate, the value of r is used as the difference interval for every element of x. The switch to central differences is indicated by C at the end of each line of intermediate printout produced by the major iterations (see Section 8.1). The use of finite differences is discussed further under the optional parameter Difference Interval.

Cold Start

Default

Warm Start

This option controls the specification of the initial working set in both the procedure for finding a feasible point for the linear constraints and bounds, and in the first QP subproblem thereafter. With a Cold Start, the first working set is chosen by E04USF/E04USA based on the values of the variables and constraints at the initial point. Broadly speaking, the initial working set will include equality constraints and bounds or inequality constraints that violate or ‘nearly’ satisfy their bounds (to within Crash Tolerance).

With a Warm Start, you must set the ISTATE array and define CLAMDA and R as discussed in Section 5. ISTATE values associated with bounds and linear constraints determine the initial working set of the procedure to find a feasible point with respect to the bounds and linear constraints. ISTATE values associated with nonlinear constraints determine the initial working set of the first QP subproblem after such a feasible point has been found. E04USF/E04USA will override your specification of ISTATE if necessary, so that a poor choice of the working set will not cause a fatal error. For instance, any elements of ISTATE which are set to -2, -1 or 4 will be reset to zero, as will any elements which are set to 3 when the corresponding elements of BL and BU are not equal. A Warm Start will be advantageous if a good estimate of the initial working set is available – for example, when E04USF/E04USA is called repeatedly to solve related problems.

Crash Tolerance

Default =0.01

This value is used in conjunction with the optional parameter Cold Start (the default value) when E04USF/E04USA selects an initial working set. If 0≤r≤1, the initial working set will include (if possible) bounds or general inequality constraints that lie within r of their bounds. In particular, a constraint of the form ajT x≥l will be included in the initial working set if ajT x-l ≤ r 1+l . If r<0 or r>1, the default value is used.

Defaults

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

Derivative Level

Default =3

This parameter indicates which derivatives are provided in user-supplied subroutines OBJFUN and CONFUN. The possible choices for i are the following.

i	Meaning
3	All elements of the objective Jacobian and the constraint Jacobian are provided by you.
2	All elements of the constraint Jacobian are provided, but some elements of the objective Jacobian are not specified by you.
1	All elements of the objective Jacobian are provided, but some elements of the constraint Jacobian are not specified by you.
0	Some elements of both the objective Jacobian and the constraint Jacobian are not specified by you.

The value i=3 should be used whenever possible, since E04USF/E04USA is more reliable (and will usually be more efficient) when all derivatives are exact.

If i=0 or 2, E04USF/E04USA will approximate unspecified elements of the objective Jacobian, using finite differences. The computation of finite difference approximations usually increases the total run-time, since a call to OBJFUN is required for each unspecified element. Furthermore, less accuracy can be attained in the solution (see Chapter 8 of Gill et al. (1981), for a discussion of limiting accuracy).

If i=0 or 1, E04USF/E04USA will approximate unspecified elements of the constraint Jacobian. One call to CONFUN is needed for each variable for which partial derivatives are not available. For example, if the constraint Jacobian has the form

* * * * * ? ? * * * ? * * * * *

where ‘*’ indicates an element provided by you and ‘?’ indicates an unspecified element, E04USF/E04USA will call CONFUN twice: once to estimate the missing element in column 2, and again to estimate the two missing elements in column 3. (Since columns 1 and 4 are known, they require no calls to CONFUN.)

At times, central differences are used rather than forward differences, in which case twice as many calls to OBJFUN and CONFUN are needed. (The switch to central differences is not under your control.)

If i<0 or i>3, the default value is used.

Difference Interval

Default values are computed

This option defines an interval used to estimate derivatives by finite differences in the following circumstances:

(a)	For verifying the objective and/or constraint gradients (see the description of the optional parameter Verify).
(b)	For estimating unspecified elements of the objective and/or constraint Jacobian matrix.

In general, a derivative with respect to the jth variable is approximated using the interval δj, where δj=r1+x^j, with x^ the first point feasible with respect to the bounds and linear constraints. If the functions are well scaled, the resulting derivative approximation should be accurate to Or. See Gill et al. (1981) for a discussion of the accuracy in finite difference approximations.

If a difference interval is not specified by you, a finite difference interval will be computed automatically for each variable by a procedure that requires up to six calls of CONFUN and OBJFUN for each element. This option is recommended if the function is badly scaled or you wish to have E04USF/E04USA determine constant elements in the objective and constraint gradients (see the descriptions of CONFUN and OBJFUN in Section 5).

Feasibility Tolerance

Default =ε

The scalar r defines the maximum acceptable absolute violations in linear and nonlinear constraints at a ‘feasible’ point; i.e., a constraint is considered satisfied if its violation does not exceed r. If r<ε or r≥1, the default value is used. Using this keyword sets both optional parameters Linear Feasibility Tolerance and Nonlinear Feasibility Tolerance to r, if ε≤r<1. (Additional details are given under the descriptions of these optional parameters.)

Function Precision

Default =ε0.9

This parameter defines εR, which is intended to be a measure of the accuracy with which the problem functions Fx and cx 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 E04USF/E04USA will not attempt to distinguish between function values that differ by less than the error inherent in the calculation.

Hessian

Default =No

This option controls the contents of the upper triangular matrix R (see Section 5). E04USF/E04USA works exclusively with the transformed and reordered Hessian HQ, and hence extra computation is required to form the Hessian itself. If Hessian=No, R contains the Cholesky factor of the transformed and reordered Hessian. If Hessian=Yes, the Cholesky factor of the approximate Hessian itself is formed and stored in R. You should select Hessian=Yes if a Warm Start will be used for the next call to E04USF/E04USA.

Infinite Bound Size

Default =1020

If r>0, r defines the ‘infinite’ bound infbnd in the definition of the problem constraints. Any upper bound greater than or equal to infbnd will be regarded as +∞ (and similarly any lower bound less than or equal to -infbnd will be regarded as -∞). If r<0, the default value is used.

Infinite Step Size

Default =maxbigbnd,1020

If r>0, r specifies the magnitude of the change in variables that is treated as a step to an unbounded solution. If the change in x during an iteration would exceed the value of r, the objective function is considered to be unbounded below in the feasible region. If r≤0, the default value is used.

JTJ Initial Hessian

Default

Unit Initial Hessian

This option controls the initial value of the upper triangular matrix R. If J denotes the objective Jacobian matrix ∇fx, then JTJ is often a good approximation to the objective Hessian matrix ∇2Fx (see also optional parameter Reset Frequency).

Line Search Tolerance

Default =0.9

The value r (0≤r<1) controls the accuracy with which the step α taken during each iteration approximates a minimum of the merit 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, particularly those with any nonlinear constraints.

If there are no nonlinear constraints, a more accurate search may be appropriate when it is desirable to reduce the number of major iterations – for example, if the objective function is cheap to evaluate, or if a substantial number of derivatives are unspecified. If r<0 or r≥1, the default value is used.

Linear Feasibility Tolerance

r₁

Default =ε

Nonlinear Feasibility Tolerance

r₂

Default =ε0.33 or ε

The default value of r2 is ε0.33 if Derivative Level=0 or 1, and ε otherwise.

The scalars r1 and r2 define the maximum acceptable absolute violations in linear and nonlinear constraints at a ‘feasible’ point; i.e., a linear constraint is considered satisfied if its violation does not exceed r1, and similarly for a nonlinear constraint and r2. If rm<ε or rm≥1, the default value is used, for m=1,2.

On entry to E04USF/E04USA, an iterative procedure is executed in order to find a point that satisfies the linear constraints and bounds on the variables to within the tolerance r1. All subsequent iterates will satisfy the linear constraints to within the same tolerance (unless r1 is comparable to the finite difference interval).

For nonlinear constraints, the feasibility tolerance r2 defines the largest constraint violation that is acceptable at an optimal point. Since nonlinear constraints are generally not satisfied until the final iterate, the value of optional parameter Nonlinear Feasibility Tolerance acts as a partial termination criterion for the iterative sequence generated by E04USF/E04USA (see also optional parameter Optimality Tolerance).

These tolerances should reflect the precision of the corresponding constraints. For example, if the variables and the coefficients in the linear constraints are of order unity, and the latter are correct to about 6 decimal digits, it would be appropriate to specify r1 as 10-6.

List

Default for E04USF=List

Nolist

Default for E04USA=Nolist

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.

Major Iteration Limit

Default =max50,3n+nL+10nN

Iteration Limit

Iters

Itns

The value of i specifies the maximum number of major iterations allowed before termination. Setting i=0 and Major Print Level>0 means that the workspace needed will be computed and printed, but no iterations will be performed. If i<0, the default value is used.

Major Print Level

Default for E04USF =10

Print Level

Default for E04USA =0

The value of i controls the amount of printout produced by the major iterations of E04USF/E04USA, as indicated below. A detailed description of the printed output is given in Section 8.1 (summary output at each major iteration and the final solution) and Section 12 (monitoring information at each major iteration). (See also the description of the optional parameter Minor Print Level.)

The following printout is sent to the current advisory message unit (as defined by X04ABF):

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

The following printout is sent to the logical unit number defined by the optional parameter Monitoring File:

i	Output
<5	No output.
≥5	One long line of output (>80 characters; see Section 12) for each major iteration (no printout of the final solution).
≥20	At each major iteration, the objective function, the Euclidean norm of the nonlinear constraint violations, the values of the nonlinear constraints (the vector c), the values of the linear constraints (the vector ALx), and the current values of the variables (the vector x).
≥30	At each major iteration, the diagonal elements of the matrix T associated with the TQ factorization (see (5) in E04UFF/E04UFA) of the QP working set, and the diagonal elements of R, the triangular factor of the transformed and reordered Hessian (see (6) in E04UFF/E04UFA).

If Major Print Level≥5 and the unit number defined by the optional parameter Monitoring File is the same as that defined by X04ABF, then the summary output for each major iteration is suppressed.

Minor Iteration Limit

Default =max50,3n+nL+nN

The value of i specifies the maximum number of iterations for finding a feasible point with respect to the bounds and linear constraints (if any). The value of i also specifies the maximum number of minor iterations for the optimality phase of each QP subproblem. If i≤0, the default value is used.

Minor Print Level

Default =0

The value of i controls the amount of printout produced by the minor iterations of E04USF/E04USA (i.e., the iterations of the quadratic programming algorithm), as indicated below. A detailed description of the printed output is given in Section 8.1 (summary output at each minor iteration and the final QP solution) and Section 12 (monitoring information at each minor iteration). (See also the description of the optional parameter Major Print Level.)

The following printout is sent to the current advisory message unit (as defined by X04ABF):

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

The following printout is sent to the logical unit number defined by the optional parameter Monitoring File:

i	Output
<5	No output.
≥5	One long line of output (>80 characters; see Section 12) for each minor iteration (no printout of the final QP solution).
≥20	At each minor iteration, the current estimates of the QP multipliers, the current estimate of the QP search direction, the QP constraint values, and the status of each QP constraint.
≥30	At each minor iteration, the diagonal elements of the matrix T associated with the TQ factorization (see (5) in E04UFF/E04UFA) of the QP working set, and the diagonal elements of the Cholesky factor R of the transformed Hessian (see (6) in E04UFF/E04UFA).

If Minor Print Level≥5 and the unit number defined by the optional parameter Monitoring File is the same as that defined by X04ABF, then the summary output for each minor iteration is suppressed.

Monitoring File

Default =-1

If i≥0 and Major Print Level≥5 or i≥0 and Minor Print Level≥5, monitoring information produced by E04USF/E04USA at every iteration is sent to a file with logical unit number i. If i<0 and/or Major Print Level<5 and Minor Print Level<5, no monitoring information is produced.

Optimality Tolerance

Default =εR0.8

The parameter r (εR≤r<1) 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 E04USF/E04USA terminates successfully, the final value of F should have approximately six correct figures. If r<εR or r≥1, the default value is used.

E04USF/E04USA will terminate successfully if the iterative sequence of x values is judged to have converged and the final point satisfies the first-order Kuhn–Tucker conditions (see Section 10.1 in E04UFF/E04UFA). The sequence of iterates is considered to have converged at x if

αp≤r1+x,

(2)

where p is the search direction and α the step length. An iterate is considered to satisfy the first-order conditions for a minimum if

ZT g FR ≤ r 1 + max 1 + F x , g FR

where ZTgFR is the projected gradient, gFR is the gradient of Fx with respect to the free variables, resj is the violation of the jth active nonlinear constraint, and ftol is the Nonlinear Feasibility Tolerance.

Reset Frequency

Default =2

If i>0, this parameter allows you to reset the approximate Hessian matrix to JTJ every i iterations, where J is the objective Jacobian matrix ∇fx (see also the description of the optional parameter JTJ Initial Hessian).

At any point where there are no nonlinear constraints active and the values of f are small in magnitude compared to the norm of J, JTJ will be a good approximation to the objective Hessian ∇2Fx. Under these circumstances, frequent resetting can significantly improve the convergence rate of E04USF/E04USA.

Resetting is suppressed at any iteration during which there are nonlinear constraints active.

If i≤0, the default value is used.

Start Objective Check At Variable

i₁

Default =1

Stop Objective Check At Variable

i₂

Default =n

Start Constraint Check At Variable

i₃

Default =1

Stop Constraint Check At Variable

i₄

Default =n

These keywords take effect only if Verify Level>0. They may be used to control the verification of Jacobian elements computed by user-supplied subroutines OBJFUN and CONFUN. For example, if the first 30 columns of the objective Jacobian appeared to be correct in an earlier run, so that only column 31 remains questionable, it is reasonable to specify Start Objective Check At Variable=31. If the first 30 variables appear linearly in the subfunctions, so that the corresponding Jacobian elements are constant, the above choice would also be appropriate.

If i2m-1≤0 or i2m-1>minn,i2m, the default value is used, for m=1 or 2. If i2m≤0 or i2m>n, the default value is used, for m=1 or 2.

Step Limit

Default =2.0

If r>0,r specifies the maximum change in variables at the first step of the linesearch. In some cases, such as Fx=aebx or Fx=axb, even a moderate change in the elements of x can lead to floating-point overflow. The parameter r is therefore used to encourage evaluation of the problem functions at meaningful points. Given any major iterate x, the first point x~ at which F and c are evaluated during the linesearch is restricted so that

x~-x2≤r1+x2.

The linesearch may go on and evaluate F and c at points further from x if this will result in a lower value of the merit function (indicated by L at the end of each line of output produced by the major iterations; see Section 8.1). If L is printed for most of the iterations, r should be set to a larger value.

Wherever possible, upper and lower bounds on x should be used to prevent evaluation of nonlinear functions at wild values. The default value Step Limit=2.0 should not affect progress on well-behaved functions, but values such as 0.1 or 0.01 may be helpful when rapidly varying functions are present. If a small value of Step Limit is selected, a good starting point may be required. An important application is to the class of nonlinear least-squares problems. If r≤0, the default value is used.

Verify Level

Default =0

Verify

Verify Constraint Gradients

Verify Gradients

Verify Objective Gradients

These keywords refer to finite difference checks on the gradient elements computed by OBJFUN and CONFUN. (Unspecified gradient elements are not checked.) The possible choices for i are the following:

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

For example, the nonlinear objective gradient (if any) will be verified if either Verify Objective Gradients or Verify Level=1 is specified. Similarly, the objective and the constraint gradients will be verified if Verify=Yes or Verify Level=3 or Verify is specified.

If i=-1, no checking will be performed.

If 0≤i≤3, gradients will be verified at the first point that satisfies the linear constraints and bounds. If i=0, only a ‘cheap’ test will be performed, requiring one call to OBJFUN and (if appropriate) one call to CONFUN. If 1≤i≤3, a more reliable (but more expensive) check will be made on individual gradient elements, within the ranges specified by the Start Objective Check At Variable and Stop Objective Check At Variable keywords. A result of the form OK or BAD? is printed by E04USF/E04USA to indicate whether or not each element appears to be correct.

If 10≤i≤13, the action is the same as for i-10, except that it will take place at the user-specified initial value of x.

If i<-1 or 4≤i≤9 or i>13, the default value is used.

We suggest that Verify Level=3 be used whenever a new function routine is being developed.

12 Description of Monitoring Information

This section describes the long line of output (>80 characters) which forms part of the monitoring information produced by E04USF/E04USA. (See also the description of the optional parameters Major Print Level, Minor Print Level and Monitoring File.) You can control the level of printed output.

When Major Print Level≥5 and Monitoring File≥0, the following line of output is produced at every major iteration of E04USF/E04USA on the unit number specified by optional parameter Monitoring File. In all cases, the values of the quantities printed are those in effect on completion of the given iteration.

Maj

is the major iteration count.

Mnr

is the number of minor iterations required by the feasibility and optimality phases of the QP subproblem. Generally, Mnr will be 1 in the later iterations, since theoretical analysis predicts that the correct active set will be identified near the solution (see Section 10 in E04UFF/E04UFA).

Note that Mnr may be greater than the optional parameter Minor Iteration Limit if some iterations are required for the feasibility phase.

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 cumulative number of evaluations of the objective function needed for the linesearch. Evaluations needed for the estimation of the gradients by finite differences are not included. Nfun is printed as a guide to the amount of work required for the linesearch.

Merit Function

is the value of the augmented Lagrangian merit function (see (12) in E04UFF/E04UFA) at the current iterate. This function will decrease at each iteration unless it was necessary to increase the penalty parameters (see Section 10.3 in E04UFF/E04UFA). As the solution is approached, Merit Function will converge to the value of the objective function at the solution.

If the QP subproblem does not have a feasible point (signified by I at the end of the current output line) then the merit function is a large multiple of the constraint violations, weighted by the penalty parameters. During a sequence of major iterations with infeasible subproblems, the sequence of Merit Function values will decrease monotonically until either a feasible subproblem is obtained or E04USF/E04USA terminates with IFAIL=3 (no feasible point could be found for the nonlinear constraints).

If there are no nonlinear constraints present (i.e., NCNLN=0) then this entry contains Objective, the value of the objective function Fx. The objective function will decrease monotonically to its optimal value when there are no nonlinear constraints.

Norm Gz

is ZTgFR, the Euclidean norm of the projected gradient (see Section 10.2 in E04UFF/E04UFA). Norm Gz will be approximately zero in the neighbourhood of a solution.

Violtn

is the Euclidean norm of the residuals of constraints that are violated or in the predicted active set (not printed if NCNLN is zero). Violtn will be approximately zero in the neighbourhood of a solution.

is the number of columns of Z (see Section 10.2 in E04UFF/E04UFA). The value of Nz is the number of variables minus the number of constraints in the predicted active set; i.e., Nz=n-Bnd+Lin+Nln.

Bnd

is the number of simple bound constraints in the current working set.

Lin

is the number of general linear constraints in the current working set.

Nln

is the number of nonlinear constraints in the predicted active set (not printed if NCNLN is zero).

Penalty

is the Euclidean norm of the vector of penalty parameters used in the augmented Lagrangian merit function (not printed if NCNLN is zero).

Cond H

is a lower bound on the condition number of the Hessian approximation H.

Cond Hz

is a lower bound on the condition number of the projected Hessian approximation HZ ( HZ = ZT HFR Z = RZT RZ ; see (6) and (11) in E04UFF/E04UFA). The larger this number, the more difficult the problem.

Cond T

is a lower bound on the condition number of the matrix of predicted active constraints.

Conv

is a three-letter indication of the status of the three convergence tests (2)–(4) defined in the description of the optional parameter Optimality Tolerance. Each letter is T if the test is satisfied and F otherwise. The three tests indicate whether:

(i)	the sequence of iterates has converged;
(ii)	the projected gradient (Norm Gz) is sufficiently small; and
(iii)	the norm of the residuals of constraints in the predicted active set (Violtn) is small enough.

If any of these indicators is F when E04USF/E04USA terminates with IFAIL=0, you should check the solution carefully.

is printed if the quasi-Newton update has been modified to ensure that the Hessian approximation is positive-definite (see Section 10.4 in E04UFF/E04UFA).

is printed if the QP subproblem has no feasible point.

is printed if central differences have been used to compute the unspecified objective and constraint gradients. If the value of Step is zero then the switch to central differences was made because no lower point could be found in the linesearch. (In this case, the QP subproblem is resolved with the central difference gradient and Jacobian.) If the value of Step is nonzero then central differences were computed because Norm Gz and Violtn imply that x is close to a Kuhn–Tucker point (see Section 10.1 in E04UFF/E04UFA).

is printed if the linesearch has produced a relative change in x greater than the value defined by the optional parameter Step Limit. If this output occurs frequently during later iterations of the run, optional parameter Step Limit should be set to a larger value.

is printed if the approximate Hessian has been refactorized. If the diagonal condition estimator of R indicates that the approximate Hessian is badly conditioned then the approximate Hessian is refactorized using column interchanges. If necessary, R is modified so that its diagonal condition estimator is bounded.

E04USF/E04USA (PDF version)

E04 Chapter Contents

E04 Chapter Introduction

NAG Library Manual

NAG Library Routine DocumentE04USF/E04USA

+− Contents

NAG Library Routine Document

E04USF/E04USA