# NAG CL Interfacee04vjc (nlp2_​sparse_​jacobian)

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## 1Purpose

e04vjc may be used before e04vhc to determine the sparsity pattern for the Jacobian.

## 2Specification

 #include
void  e04vjc (Integer nf, Integer n,
 void (*usrfun)(Integer *status, Integer n, const double x[], Integer needf, Integer nf, double f[], Integer needg, Integer leng, double g[], Nag_Comm *comm),
Integer iafun[], Integer javar[], double a[], Integer lena, Integer *nea, Integer igfun[], Integer jgvar[], Integer leng, Integer *neg, const double x[], const double xlow[], const double xupp[], Nag_E04State *state, Nag_Comm *comm, NagError *fail)
The function may be called by the names: e04vjc, nag_opt_nlp2_sparse_jacobian or nag_opt_sparse_nlp_jacobian.

## 3Description

When using e04vhc, if you set the optional parameter ${\mathbf{Derivative Option}}=0$ and usrfun provides none of the derivatives, you may need to call e04vjc to determine the input arrays iafun, javar, a, igfun and jgvar. These arrays define the pattern of nonzeros in the Jacobian matrix. A typical sequence of calls could be
```e04vgc (&state, ... );
e04vjc (nf, n, ... );
e04vlc ("Derivative Option = 0", &state, ... );
e04vhc (start, nf, ... );```
e04vjc determines the sparsity pattern for the Jacobian and identifies the constant elements automatically. To do so, e04vjc approximates the problem functions, $F\left(x\right)$, at three random perturbations of the given initial point $x$. If an element of the approximate Jacobian is the same at all three points, then it is taken to be constant. If it is zero, it is taken to be identically zero. Since the random points are not chosen close together, the heuristic will correctly classify the Jacobian elements in the vast majority of cases. In general, e04vjc finds that the Jacobian can be permuted to the form:
 $( G(x) A3 A2 A4 ) ,$
where ${A}_{2}$, ${A}_{3}$ and ${A}_{4}$ are constant. Note that $G\left(x\right)$ might contain elements that are also constant, but e04vjc must classify them as nonlinear. This is because e04vhc ‘removes’ linear variables from the calculation of $F$ by setting them to zero before calling usrfun. A knowledgeable user would be able to move such elements from $F\left(x\right)$ in usrfun and enter them as part of iafun, javar and a for e04vhc.
Hock W and Schittkowski K (1981) Test Examples for Nonlinear Programming Codes. Lecture Notes in Economics and Mathematical Systems 187 Springer–Verlag

## 5Arguments

Note: all optional parameters are described in detail in Section 12.1 in e04vhc.
1: $\mathbf{nf}$Integer Input
On entry: $\mathit{nf}$, the number of problem functions in $F\left(x\right)$, including the objective function (if any) and the linear and nonlinear constraints. Simple upper and lower bounds on $x$ can be defined using the arguments xlow and xupp and should not be included in $F$.
Constraint: ${\mathbf{nf}}>0$.
2: $\mathbf{n}$Integer Input
On entry: $n$, the number of variables.
Constraint: ${\mathbf{n}}>0$.
3: $\mathbf{usrfun}$function, supplied by the user External Function
usrfun must define the problem functions $F\left(x\right)$. This function is passed to e04vjc as the external argument usrfun.
The specification of usrfun is:
 void usrfun (Integer *status, Integer n, const double x[], Integer needf, Integer nf, double f[], Integer needg, Integer leng, double g[], Nag_Comm *comm)
1: $\mathbf{status}$Integer * Input/Output
On entry: indicates the first call to usrfun.
${\mathbf{status}}=0$
There is nothing special about the current call to usrfun.
${\mathbf{status}}=1$
e04vjc is calling your function for the first time. Some data may need to be input or computed and saved.
On exit: may be used to indicate that you are unable to evaluate $F$ at the current $x$. (For example, the problem functions may not be defined there).
e04vjc evaluates $F\left(x\right)$ at random perturbation of the initial point $x$, say ${x}_{p}$. If the functions cannot be evaluated at ${x}_{p}$, you can set ${\mathbf{status}}=-1$, e04vjc will use another random perturbation.
If for some reason you wish to terminate the current problem, set ${\mathbf{status}}\le -2$.
2: $\mathbf{n}$Integer Input
On entry: $n$, the number of variables, as defined in the call to e04vjc.
3: $\mathbf{x}\left[{\mathbf{n}}\right]$const double Input
On entry: the variables $x$ at which the problem functions are to be calculated. The array $x$ must not be altered.
4: $\mathbf{needf}$Integer Input
On entry: indicates if f must be assigned during the call to usrfun (see f).
5: $\mathbf{nf}$Integer Input
On entry: $\mathit{nf}$, the number of problem functions.
6: $\mathbf{f}\left[{\mathbf{nf}}\right]$double Input/Output
On entry: this will be set by e04vjc.
On exit: the computed $F\left(x\right)$ according to the setting of needf.
If ${\mathbf{needf}}=0$, f is not required and is ignored.
If ${\mathbf{needf}}>0$, the components of $F\left(x\right)$ must be calculated and assigned to f. e04vjc will always call usrfun with ${\mathbf{needf}}>0$.
To simplify the code, you may ignore the value of needf and compute $F\left(x\right)$ on every entry to usrfun.
7: $\mathbf{needg}$Integer Input
On entry: e04vjc will call usrfun with ${\mathbf{needg}}=0$ to indicate that g is not required.
8: $\mathbf{leng}$Integer Input
On entry: the dimension of the array g.
9: $\mathbf{g}\left[{\mathbf{leng}}\right]$double Input/Output
On entry: concerns the calculations of the derivatives of the function $f\left(x\right)$.
On exit: e04vjc will always call usrfun with ${\mathbf{needg}}=0$: g is not required to be set on exit but must be declared correctly.
10: $\mathbf{comm}$Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to usrfun.
userdouble *
iuserInteger *
pPointer
The type Pointer will be void *. Before calling e04vjc you may allocate memory and initialize these pointers with various quantities for use by usrfun when called from e04vjc (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
Note: usrfun should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by e04vjc. If your code inadvertently does return any NaNs or infinities, e04vjc is likely to produce unexpected results.
4: $\mathbf{iafun}\left[{\mathbf{lena}}\right]$Integer Output
5: $\mathbf{javar}\left[{\mathbf{lena}}\right]$Integer Output
6: $\mathbf{a}\left[{\mathbf{lena}}\right]$double Output
On exit: define the coordinates $\left(i,j\right)$ and values ${A}_{ij}$ of the nonzero elements of the linear part $A$ of the function $F\left(x\right)=f\left(x\right)+Ax$.
In particular, nea triples $\left({\mathbf{iafun}}\left[k-1\right],{\mathbf{javar}}\left[k-1\right],{\mathbf{a}}\left[k-1\right]\right)$ define the row and column indices $i={\mathbf{iafun}}\left[k-1\right]$ and $j={\mathbf{javar}}\left[k-1\right]$ of the element ${A}_{ij}={\mathbf{a}}\left[k-1\right]$.
7: $\mathbf{lena}$Integer Input
On entry: the dimension of the arrays iafun, javar and a that hold $\left(i,j,{A}_{ij}\right)$. lena should be an overestimate of the number of elements in the linear part of the Jacobian.
Constraint: ${\mathbf{lena}}\ge 1$.
8: $\mathbf{nea}$Integer * Output
On exit: is the number of nonzero entries in $A$ such that $F\left(x\right)=f\left(x\right)+Ax$.
9: $\mathbf{igfun}\left[{\mathbf{leng}}\right]$Integer Output
10: $\mathbf{jgvar}\left[{\mathbf{leng}}\right]$Integer Output
On exit: define the coordinates $\left(i,j\right)$ of the nonzero elements of $G$, the nonlinear part of the derivatives $J\left(x\right)=G\left(x\right)+A$ of the function $F\left(x\right)=f\left(x\right)+Ax$.
11: $\mathbf{leng}$Integer Input
On entry: the dimension of the arrays igfun and jgvar that define the varying Jacobian elements $\left(i,j,{G}_{ij}\right)$. leng should be an overestimate of the number of elements in the nonlinear part of the Jacobian.
Constraint: ${\mathbf{leng}}\ge 1$.
12: $\mathbf{neg}$Integer * Output
On exit: the number of nonzero entries in $G$.
13: $\mathbf{x}\left[{\mathbf{n}}\right]$const double Input
On entry: an initial estimate of the variables $x$. The contents of $x$ will be used by e04vjc in the call of usrfun, and so each element of x should be within the bounds given by xlow and xupp.
14: $\mathbf{xlow}\left[{\mathbf{n}}\right]$const double Input
15: $\mathbf{xupp}\left[{\mathbf{n}}\right]$const double Input
On entry: contain the lower and upper bounds ${l}_{x}$ and ${u}_{x}$ on the variables $x$.
To specify a nonexistent lower bound ${\left[{l}_{x}\right]}_{j}=-\infty$, set ${\mathbf{xlow}}\left[j-1\right]\le -\mathit{bigbnd}$, where $\mathit{bigbnd}$ is the optional parameter ${\mathbf{Infinite Bound Size}}$. To specify a nonexistent upper bound ${\mathbf{xupp}}\left[j-1\right]\ge \mathit{bigbnd}$.
To fix the $j$th variable (say, ${x}_{j}=\beta$, where $|\beta |<\mathit{bigbnd}$), set ${\mathbf{xlow}}\left[j-1\right]={\mathbf{xupp}}\left[j-1\right]=\beta$.
16: $\mathbf{state}$Nag_E04State * Communication Structure
state contains internal information required for functions in this suite. It must not be modified in any way.
17: $\mathbf{comm}$Nag_Comm *
The NAG communication argument (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
18: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
Internal error: memory allocation failed when attempting to allocate workspace sizes $⟨\mathit{\text{value}}⟩$, $⟨\mathit{\text{value}}⟩$ and $⟨\mathit{\text{value}}⟩$. Please contact NAG.
NE_ALLOC_INSUFFICIENT
NE_ARRAY_TOO_SMALL
Either lena or leng is too small. Increase both of them and corresponding array sizes. ${\mathbf{lena}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{leng}}=⟨\mathit{\text{value}}⟩$.
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_E04VGC_NOT_INIT
The initialization function e04vgc has not been called.
NE_INT
On entry, ${\mathbf{lena}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{lena}}\ge 1$.
On entry, ${\mathbf{leng}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{leng}}\ge 1$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_JACOBIAN_STRUCTURE_FAIL
Cannot estimate Jacobian structure at given point x.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_USER_STOP
User-supplied function usrfun requested termination.
You have indicated the wish to terminate the call to e04vjc by setting status to a value $\text{}<-1$ on exit from usrfun.
NE_USRFUN_UNDEFINED
User-supplied function usrfun indicates that functions are undefined near given point x.
You have indicated that the problem functions are undefined by setting ${\mathbf{status}}=-1$ on exit from usrfun. This exit occurs if e04vjc is unable to find a point at which the functions are defined.

Not applicable.

## 8Parallelism and Performance

e04vjc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

None.

## 10Example

This example shows how to call e04vjc to determine the sparsity pattern of the Jacobian before calling e04vhc to solve a sparse nonlinear programming problem without providing the Jacobian information in usrfun.
It is a reformulation of Problem 74 from Hock and Schittkowski (1981) and involves the minimization of the nonlinear function
 $f(x) = 10−6 x33 + 23 × 10−6 x43+3 x3+2 x4$
subject to the bounds
 $-0.55≤x1≤ 0.55, -0.55≤x2≤ 0.55, 0≤x3≤ 1200, 0≤x4≤ 1200,$
to the nonlinear constraints
 $1000sin(-x1-0.25)+1000sin(-x2-0.25)-x3 = -894.8, 1000sin(x1-0.25)+1000sin(x1-x2-0.25)-x4 = -894.8, 1000sin(x2-0.25)+1000sin(x2-x1-0.25) = -1294.8,$
and to the linear constraints
 $-x1+x2≥-0.55, -x1-x2≥-0.55.$
The initial point, which is infeasible, is
 $x0 = ( 0, 0, 0, 0 ) T ,$
and $f\left({x}_{0}\right)=0$.
The optimal solution (to five figures) is
 $x*=(0.11887,-0.39623,679.94,1026.0)T,$
and $f\left({x}^{*}\right)=5126.4$. All the nonlinear constraints are active at the solution.
The formulation of the problem combines the constraints and the objective into a single vector ($F$).
 $F = ( 1000 sin(-x1-0.25) + 1000 sin(-x2-0.25) - x3 1000 sin(x1-0.25) + 1000 sin(x1-x2-0.25) - x4 1000 sin(x2-0.25) + 1000 sin(x2-x1-0.25) -x1 + x2 x1 - x2 10−6 x33 + 23 × 10−6 x43 + 3x3 + 2x4 )$

### 10.1Program Text

Program Text (e04vjce.c)

### 10.2Program Data

Program Data (e04vjce.d)

### 10.3Program Results

Program Results (e04vjce.r)