NAG Library Routine Document

E04USF/E04USA

1  Purpose

2  Specification

2.1  Specification for E04USF

2.2  Specification for E04USA

3  Description

4  References

5  Parameters

1:     M – INTEGERInput

On entry: $m$, the number of subfunctions associated with $F\left(x\right)$.

Constraint: $\mathbf{M}>0$.

2:     N – INTEGERInput

On entry: $n$, the number of variables.

Constraint: $\mathbf{N}>0$.

3:     NCLIN – INTEGERInput

On entry: $n_L$, the number of general linear constraints.

Constraint: $\mathbf{NCLIN}\ge 0$.

4:     NCNLN – INTEGERInput

On entry: $n_N$, the number of nonlinear constraints.

Constraint: $\mathbf{NCNLN}\ge 0$.

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: $\mathbf{LDA}\ge \max \left(1,\mathbf{NCLIN}\right)$.

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: $\mathbf{LDCJ}\ge \max \left(1,\mathbf{NCNLN}\right)$.

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: $\mathbf{LDFJ}\ge \mathbf{M}$.

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: $\mathbf{LDR}\ge \mathbf{N}$.

9:     A(LDA,$*$) – REAL (KIND=nag_wp) arrayInput

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

On entry: the $\mathit{i}$th row of A contains the $\mathit{i}$th row of the matrix $A_L$ of general linear constraints in (1). That is, the $\mathit{i}$th row contains the coefficients of the $\mathit{i}$th general linear constraint, for $\mathit{i}=1,2,\dots ,\mathbf{NCLIN}$.

If $\mathbf{NCLIN}=0$, the array A is not referenced.

10:   BL($\mathbf{N}+\mathbf{NCLIN}+\mathbf{NCNLN}$) – REAL (KIND=nag_wp) arrayInput

11:   BU($\mathbf{N}+\mathbf{NCLIN}+\mathbf{NCNLN}$) – REAL (KIND=nag_wp) 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 $n_L$ elements the bounds for the general linear constraints (if any) and the next $n_N$ elements the bounds for the general nonlinear constraints (if any). To specify a nonexistent lower bound (i.e., $l_j=-\infty$), set $\mathbf{BL}\left(j\right)\le -\mathit{bigbnd}$, and to specify a nonexistent upper bound (i.e., $u_j=+\infty$), set $\mathbf{BU}\left(j\right)\ge \mathit{bigbnd}$; the default value of $\mathit{bigbnd}$ is ${10}^{20}$, but this may be changed by the optional parameter Infinite Bound Size. To specify the $j$th constraint as an equality, set $\mathbf{BL}\left(j\right)=\mathbf{BU}\left(j\right)=\beta$, say, where $\left|\beta \right|<\mathit{bigbnd}$.

Constraints:

• $\mathbf{BL}\left(\mathit{j}\right)\le \mathbf{BU}\left(\mathit{j}\right)$, for $\mathit{j}=1,2,\dots ,\mathbf{N}+\mathbf{NCLIN}+\mathbf{NCNLN}$;
• if $\mathbf{BL}\left(j\right)=\mathbf{BU}\left(j\right)=\beta$, $\left|\beta \right|<\mathit{bigbnd}$.

12:   Y(M) – REAL (KIND=nag_wp) 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 $c\left(x\right)$ of nonlinear constraint functions and (optionally) its Jacobian ($=\frac{\partial c}{\partial x}$) for a specified $n$-element vector $x$. If there are no nonlinear constraints (i.e., $\mathbf{NCNLN}=0$), 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(*) REAL (KIND=nag_wp)  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 $\mathbf{NEEDC}\left(i\right)>0$:

$\mathbf{MODE}=0$

$\mathbf{C}\left(i\right)$.

$\mathbf{MODE}=1$

All available elements in the $i$th row of CJAC.

$\mathbf{MODE}=2$

$\mathbf{C}\left(i\right)$ and all available elements in the $i$th 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: $n_N$, 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 $\mathbf{NEEDC}\left(i\right)>0$, then the $i$th element of C and/or the available elements of the $i$th row of CJAC (see parameter MODE) must be evaluated at $x$.

6:     X(N) – REAL (KIND=nag_wp) 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) – REAL (KIND=nag_wp) arrayOutput

On exit: if $\mathbf{NEEDC}\left(i\right)>0$ and $\mathbf{MODE}=0$ or $2$, $\mathbf{C}\left(i\right)$ must contain the value of the $i$th constraint at $x$. The remaining elements of C, corresponding to the non-positive elements of NEEDC, are ignored.

8:     CJAC(LDCJ,N) – REAL (KIND=nag_wp) arrayInput/Output

On entry: is set to a special value.

On exit: if $\mathbf{NEEDC}\left(i\right)>0$ and $\mathbf{MODE}=1$ or $2$, the $i$th row of CJAC must contain the available elements of the vector $\nabla c_i$ given by

$$\nabla c_i={\left(\frac{\partial c_i}{\partial x_1},\frac{\partial c_i}{\partial x_2},\dots ,\frac{\partial c_i}{\partial x_n}\right)}^{\mathrm{T}}\text{,}$$

where $\frac{\partial c_i}{\partial x_j}$ is the partial derivative of the $i$th constraint with respect to the $j$th variable, evaluated at the point $x$. See also the parameter NSTATE. The remaining rows of CJAC, corresponding to non-positive elements of NEEDC, are ignored.

If all elements of the constraint Jacobian are known (i.e., $\mathbf{Derivative\; Level}=2$ 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 $\mathbf{NSTATE}=1$). 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 $\mathbf{CJAC}\left(i,j\right)$ is set to a constant value, it need not be reset in subsequent calls to CONFUN, but the value $\mathbf{CJAC}\left(i,j\right)\times \mathbf{X}\left(j\right)$ must nonetheless be added to $\mathbf{C}\left(i\right)$. For example, if $\mathbf{CJAC}\left(1,1\right)=2$ and $\mathbf{CJAC}\left(1,2\right)=-5$, then the term $2\times \mathbf{X}\left(1\right)-5\times \mathbf{X}\left(2\right)$ must be included in the definition of $\mathbf{C}\left(1\right)$.

It must be emphasized that, if $\mathbf{Derivative\; Level}=0$ 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 $\mathbf{NSTATE}=1$ 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($*$) – REAL (KIND=nag_wp) arrayUser Workspace

CONFUN is called 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 either be a module subprogram USEd by, or 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 $i$th element of the vector $f\left(x\right)={\left(f_1\left(x\right),f_2\left(x\right),\dots ,f_m\left(x\right)\right)}^{\mathrm{T}}$ or all $m$ elements of $f\left(x\right)$ and (optionally) its Jacobian ($=\frac{\partial f}{\partial x}$) 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(*) REAL (KIND=nag_wp)  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:

$\mathbf{MODE}=0$ and $\mathbf{NEEDFI}=i$, where $i>0$

$\mathbf{F}\left(i\right)$.

$\mathbf{MODE}=0$ and $\mathbf{NEEDFI}<0$

F.

$\mathbf{MODE}=1$ and $\mathbf{NEEDFI}<0$

All available elements of FJAC.

$\mathbf{MODE}=2$ and $\mathbf{NEEDFI}<0$

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 $\mathbf{NEEDFI}=i>0$, only the $i$th element of $f\left(x\right)$ needs to be evaluated at $x$; the remaining elements need not be set. This can result in significant computational savings when $m\gg n$.

6:     X(N) – REAL (KIND=nag_wp) arrayInput

On entry: $x$, the vector of variables at which $f\left(x\right)$ and/or all available elements of its Jacobian are to be evaluated.

7:     F(M) – REAL (KIND=nag_wp) arrayOutput

On exit: if $\mathbf{MODE}=0$ and $\mathbf{NEEDFI}=i>0$, $\mathbf{F}\left(i\right)$ must contain the value of $f_i$ at $x$.

If $\mathbf{MODE}=0$ or $2$ and $\mathbf{NEEDFI}<0$, $\mathbf{F}\left(\mathit{i}\right)$ must contain the value of $f_{\mathit{i}}$ at $x$, for $\mathit{i}=1,2,\dots ,m$.

8:     FJAC(LDFJ,N) – REAL (KIND=nag_wp) arrayInput/Output

On entry: is set to a special value.

On exit: if $\mathbf{MODE}=1$ or $2$ and $\mathbf{NEEDFI}<0$, the $i$th row of FJAC must contain the available elements of the vector $\nabla f_i$ given by

$$\nabla f_i={\left(\frac{\partial f_i}{\partial x_1},\frac{\partial f_i}{\partial x_2},\dots ,\frac{\partial f_i}{\partial x_n}\right)}^{\mathrm{T}}\text{,}$$

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

9:     NSTATE – INTEGERInput

On entry: if $\mathbf{NSTATE}=1$ 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($*$) – REAL (KIND=nag_wp) arrayUser Workspace

OBJFUN is called 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 either be a module subprogram USEd by, or 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($\mathbf{N}+\mathbf{NCLIN}+\mathbf{NCNLN}$) – 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 $n_L$ elements refer to the upper and lower bounds on $A_{L}x$, and the next $n_N$ elements refer to the upper and lower bounds on $c\left(x\right)$. Possible values for $\mathbf{ISTATE}\left(j\right)$ are as follows:

$\mathbf{ISTATE}\left(j\right)$	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 $\mathbf{BL}\left(j\right)=\mathbf{BU}\left(j\right)$.

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: $-2\le \mathbf{ISTATE}\left(\mathit{j}\right)\le 4$, for $\mathit{j}=1,2,\dots ,\mathbf{N}+\mathbf{NCLIN}+\mathbf{NCNLN}$.

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 $\mathbf{ISTATE}\left(j\right)$ is as follows:

$\mathbf{ISTATE}\left(j\right)$	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.
$\phantom{-}0$	The constraint is satisfied to within the feasibility tolerance, but is not in the QP working set.
$\phantom{-}1$	This inequality constraint is included in the QP working set at its lower bound.
$\phantom{-}2$	This inequality constraint is included in the QP working set at its upper bound.
$\phantom{-}3$	This constraint is included in the QP working set as an equality. This value of ISTATE can occur only when $\mathbf{BL}\left(j\right)=\mathbf{BU}\left(j\right)$.

17:   C($\max \left(1,\mathbf{NCNLN}\right)$) – REAL (KIND=nag_wp) arrayOutput

On exit: if $\mathbf{NCNLN}>0$, $\mathbf{C}\left(\mathit{i}\right)$ contains the value of the $\mathit{i}$th nonlinear constraint function $c_{\mathit{i}}$ at the final iterate, for $\mathit{i}=1,2,\dots ,\mathbf{NCNLN}$.

If $\mathbf{NCNLN}=0$, the array C is not referenced.

18:   CJAC(LDCJ,$*$) – REAL (KIND=nag_wp) arrayInput/Output

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

On entry: in general, CJAC need not be initialized before the call to E04USF/E04USA. However, if $\mathbf{Derivative\; Level}=3$, 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 $\mathbf{NCNLN}>0$, CJAC contains the Jacobian matrix of the nonlinear constraint functions at the final iterate, i.e., $\mathbf{CJAC}\left(\mathit{i},\mathit{j}\right)$ contains the partial derivative of the $\mathit{i}$th constraint function with respect to the $\mathit{j}$th variable, for $\mathit{i}=1,2,\dots ,\mathbf{NCNLN}$ and $\mathit{j}=1,2,\dots ,\mathbf{N}$. (See the discussion of parameter CJAC under CONFUN.)

If $\mathbf{NCNLN}=0$, the array CJAC is not referenced.

19:   F(M) – REAL (KIND=nag_wp) arrayOutput

On exit: $\mathbf{F}\left(\mathit{i}\right)$ contains the value of the $\mathit{i}$th function $f_{\mathit{i}}$ at the final iterate, for $\mathit{i}=1,2,\dots ,\mathbf{M}$.

20:   FJAC(LDFJ,N) – REAL (KIND=nag_wp) arrayInput/Output

On entry: in general, FJAC need not be initialized before the call to E04USF/E04USA. However, if $\mathbf{Derivative\; Level}=3$, 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 $f_1,f_2,\dots ,f_m$ at the final iterate, i.e., $\mathbf{FJAC}\left(\mathit{i},\mathit{j}\right)$ contains the partial derivative of the $\mathit{i}$th function with respect to the $\mathit{j}$th variable, for $\mathit{i}=1,2,\dots ,\mathbf{M}$ and $\mathit{j}=1,2,\dots ,\mathbf{N}$. (See also the discussion of parameter FJAC under OBJFUN.)

21:   CLAMDA($\mathbf{N}+\mathbf{NCLIN}+\mathbf{NCNLN}$) – REAL (KIND=nag_wp) 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, $\mathbf{CLAMDA}\left(\mathit{j}\right)$ 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 $\mathit{j}=\mathbf{N}+\mathbf{NCLIN}+1,\dots ,\mathbf{N}+\mathbf{NCLIN}+\mathbf{NCNLN}$. The remaining elements need not be set. Note that if the $j$th constraint is defined as ‘inactive’ by the initial value of the ISTATE array (i.e., $\mathbf{ISTATE}\left(j\right)=0$), $\mathbf{CLAMDA}\left(j\right)$ should be zero; if the $j$th constraint is an inequality active at its lower bound (i.e., $\mathbf{ISTATE}\left(j\right)=1$), $\mathbf{CLAMDA}\left(j\right)$ should be non-negative; if the $j$th constraint is an inequality active at its upper bound (i.e., $\mathbf{ISTATE}\left(j\right)=2$, $\mathbf{CLAMDA}\left(j\right)$ should be non-positive. If necessary, the routine will modify CLAMDA to match these rules.

On exit: the values of the QP multipliers from the last QP subproblem. $\mathbf{CLAMDA}\left(j\right)$ should be non-negative if $\mathbf{ISTATE}\left(j\right)=1$ and non-positive if $\mathbf{ISTATE}\left(j\right)=2$.

22:   OBJF – REAL (KIND=nag_wp)Output

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

23:   R(LDR,N) – REAL (KIND=nag_wp) 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 $\mathbf{Hessian}=\mathrm{NO}$, R contains the upper triangular Cholesky factor $R$ of $Q^{\mathrm{T}}\tilde{H}Q$, an estimate of the transformed and reordered Hessian of the Lagrangian at $x$ (see (6) in E04UFF/E04UFA). If $\mathbf{Hessian}=\mathrm{YES}$, 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) – REAL (KIND=nag_wp) 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: $\mathbf{LIWORK}\ge 3\times \mathbf{N}+\mathbf{NCLIN}+2\times \mathbf{NCNLN}$.

27:   WORK(LWORK) – REAL (KIND=nag_wp) 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 $\mathbf{NCNLN}=0$ and $\mathbf{NCLIN}=0$, $\mathbf{LWORK}\ge 20\times \mathbf{N}+\mathbf{M}\times \left(\mathbf{N}+3\right)$;
• if $\mathbf{NCNLN}=0$ and $\mathbf{NCLIN}>0$, $$\begin{gathered} \mathbf{LWORK}\ge 2\times {\mathbf{N}}^2+20\times \mathbf{N}+11\times \mathbf{NCLIN}+ \\ \mathbf{M}\times \left(\mathbf{N}+3\right) \end{gathered}$$;
• if $\mathbf{NCNLN}>0$ and $\mathbf{NCLIN}\ge 0$, $$\begin{gathered} \mathbf{LWORK}\ge 2\times {\mathbf{N}}^2+\mathbf{N}\times \mathbf{NCLIN}+2\times \mathbf{N}\times \\ \mathbf{NCNLN}+20\times \mathbf{N}+11\times \mathbf{NCLIN}+21\times \mathbf{NCNLN}+\mathbf{M}\times \left(\mathbf{N}+3\right) \end{gathered}$$.

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 $\mathbf{IFAIL}=\mathbf{9}$.)

29:   IUSER($*$) – INTEGER arrayUser Workspace

30:   RUSER($*$) – REAL (KIND=nag_wp) arrayUser Workspace

IUSER and RUSER are not used by E04USF/E04USA, but are passed directly to CONFUN and OBJFUN and may be used to pass information to these routines as an alternative to using COMMON global variables.

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 $0$, $-1\text{ or }1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{ or }1$ 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 $\mathbf{IFAIL}\ne \mathbf{0}$ on exit, the recommended value is $-1$. When the value $-\mathbf{1}\text{ or }1$ is used it is essential to test the value of IFAIL on exit.

On exit: $\mathbf{IFAIL}=\mathbf{0}$ unless the routine detects an error or a warning has been flagged (see Section 6).

E04USF/E04USA returns with $\mathbf{IFAIL}=\mathbf{0}$ 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 ($\text{default value}={\epsilon }_r^{0.8}$, where ${\epsilon }_r$ is the value of the optional parameter Function Precision ($\text{default value}={\epsilon }^{0.9}$, where $\epsilon$ 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($475$) – REAL (KIND=nag_wp) arrayCommunication Array

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

34:   IFAIL – INTEGERInput/Output

Note: see the parameter description for IFAIL above.

6  Error Indicators and Warnings

$\mathbf{IFAIL}<0$

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

$\mathbf{IFAIL}=1$

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 ($\text{default value}={\epsilon }^{0.9}$, where $\epsilon$ 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 $\mathbf{IFAIL}=\mathbf{0}$ are satisfied, $x$ is likely to be a solution of (1) even if $\mathbf{IFAIL}=\mathbf{1}$.

$\mathbf{IFAIL}=2$

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 ($\text{default value}=\sqrt{\epsilon }$, where $\epsilon$ is the machine precision), or no feasible point could be found in the number of iterations specified by the optional parameter Minor Iteration Limit ($\text{default value}=\max \left(50,3\left(n+n_L+n_N\right)\right)$). You should check that there are no constraint redundancies. If the data for the constraints are accurate only to an absolute precision $\sigma$, you should ensure that the value of the optional parameter Linear Feasibility Tolerance is greater than $\sigma$. For example, if all elements of $A_L$ are of order unity and are accurate to only three decimal places, Linear Feasibility Tolerance should be at least ${10}^{-3}$.

$\mathbf{IFAIL}=3$

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.

$\mathbf{IFAIL}=4$

The limiting number of iterations (as determined by the optional parameter Major Iteration Limit ($\text{default value}=\max \left(50,3\left(n+n_L\right)+10n_N\right)$) 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 $\mathbf{IFAIL}=\mathbf{6}$.

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 $\mathbf{R}$ 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.

$\mathbf{IFAIL}=5$

Not used by this routine.

$\mathbf{IFAIL}=6$

$x$ 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 ($\text{default value}={\epsilon }_r^{0.8}$, where ${\epsilon }_r$ is the value of the optional parameter Function Precision ($\text{default value}={\epsilon }^{0.9}$, where $\epsilon$ is the machine precision)) is too small. In this case you should apply the four tests described under $\mathbf{IFAIL}=\mathbf{0}$ 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 $\mathbf{IFAIL}=\mathbf{7}$ and check the gradients using the optional parameter Verify ($\text{default value}=0$). 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 $\mathbf{R}$ 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 ($\text{default value}=\sqrt{\epsilon }$, where $\epsilon$ 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 $\mathbf{Major\; Print\; Level}\ge 30$).

$\mathbf{IFAIL}=7$

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 $F\left(x\right)$ 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.

$\mathbf{IFAIL}=8$

Not used by this routine.

$\mathbf{IFAIL}=9$

An input parameter is invalid.

$\mathbf{\text{overflow}}$

If overflow occurs then either an element of $C$ is very large, or the singular values or singular vectors have been incorrectly supplied.

7  Accuracy

8  Further Comments

8.1  Description of the Printed Output

9  Example

9.1  Program Text

Program Text (e04usfe.f90)

Program Text (e04usae.f90)

9.2  Program Data

Program Data (e04usfe.d)

Program Data (e04usae.d)

9.3  Program Results

Program Results (e04usfe.r)

Program Results (e04usae.r)

10  Algorithmic Details

11  Optional Parameters

• Central Difference Interval
• Cold Start
• Crash Tolerance
• Defaults
• Derivative Level
• Difference Interval
• Feasibility Tolerance
• Function Precision
• Hessian
• Infinite Bound Size
• Infinite Step Size
• Iteration Limit
• Iters
• Itns
• JTJ Initial Hessian
• Linear Feasibility Tolerance
• Line Search Tolerance
• List
• Major Iteration Limit
• Major Print Level
• Minor Iteration Limit
• Minor Print Level
• Monitoring File
• Nolist
• Nonlinear Feasibility Tolerance
• Optimality Tolerance
• Print Level
• Reset Frequency
• Start Constraint Check At Variable
• Start Objective Check At Variable
• Step Limit
• Stop Constraint Check At Variable
• Stop Objective Check At Variable
• Unit Initial Hessian
• Verify
• Verify Constraint Gradients
• Verify Gradients
• Verify Level
• Verify Objective Gradients
• Warm Start

Begin
  Print level = 1
End

 CALL E04UQF (IOPTNS, INFORM)

 CALL E04URF ('Print Level = 1')

11.1  Description of the Optional Parameters

• the keywords, where the minimum abbreviation of each keyword is underlined (if no characters of an optional qualifier are underlined, the qualifier may be omitted);
• a parameter value, where the letters $a$, $i\text{ and }r$ denote options that take character, integer and real values respectively;
• the default value, where the symbol $\epsilon$ is a generic notation for machine precision (see X02AJF), and ${\epsilon }_r$ denotes the relative precision of the objective function Function Precision.

12  Description of Monitoring Information