NAG Library Function Document

nag_opt_bounds_qa_no_deriv (e04jcc)

1  Purpose

2  Specification

3  Description

4  References

5  Arguments

1:     objfun – function, supplied by the userExternal Function

objfun must evaluate the objective function $F$ at a specified vector $\mathbf{x}$.

The specification of objfun is:

void  objfun (Integer n, const double x[], double *f, Nag_Comm *comm, Integer *inform)

1:     n – IntegerInput

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

2:     x[n] – const doubleInput

On entry: $\mathbf{x}$, the vector at which the objective function is to be evaluated.

3:     f – double *Output

On exit: must be set to the value of the objective function at $\mathbf{x}$, unless you have specified termination of the current problem using inform.

4:     comm – Nag_Comm *

Pointer to structure of type Nag_Comm; the following members are relevant to objfun.

user – double *

iuser – Integer *

p – Pointer 

The type Pointer will be void *. Before calling nag_opt_bounds_qa_no_deriv (e04jcc) you may allocate memory and initialize these pointers with various quantities for use by objfun when called from nag_opt_bounds_qa_no_deriv (e04jcc) (see Section 3.2.1.1 in the Essential Introduction).

5:     inform – Integer *Output

On exit: must be set to a value describing the action to be taken by the solver on return from objfun. Specifically, if the value is negative the solution of the current problem will terminate immediately; otherwise, computations will continue.

2:     n – IntegerInput

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

Constraint: $\mathbf{n}\ge 2$ and $n_r\ge 2$, where $n_r$ denotes the number of non-fixed variables.

3:     npt – IntegerInput

On entry: $m$, the number of interpolation conditions imposed on the quadratic approximation at each iteration.

Suggested value: $\mathbf{npt}=2\times n_r+1$, where $n_r$ denotes the number of non-fixed variables.

Constraint: $n_r+2\le \mathbf{npt}\le \frac{\left(n_r+1\right)\times \left(n_r+2\right)}{2}$, where $n_r$ denotes the number of non-fixed variables.

4:     x[n] – doubleInput/Output

On entry: an estimate of the position of the minimum. If any component is out-of-bounds it is replaced internally by the bound it violates.

On exit: the lowest point found during the calculations. Thus, if $\mathbf{fail}\mathbf{.}\mathbf{code}=$ NE_NOERROR on exit, x is the position of the minimum.

5:     bl[n] – const doubleInput

6:     bu[n] – const doubleInput

On entry: the fixed vectors of bounds: the lower bounds $\mathbf{\ell }$ and the upper bounds $\mathbf{u}$, respectively. To signify that a variable is unbounded you should choose a large scalar $r$ appropriate to your problem, then set the lower bound on that variable to $-r$ and the upper bound to $r$. For well-scaled problems $r=r_{\max }^{\frac{1}{4}}$ may be suitable, where $r_{\max }$ denotes the largest positive model number (see nag_real_largest_number (X02ALC)).

Constraints:

• if $\mathbf{x}\left[\mathit{i}-1\right]$ is to be fixed at $\mathbf{bl}\left[\mathit{i}-1\right]$, then $\mathbf{bl}\left[\mathit{i}-1\right]=\mathbf{bu}\left[\mathit{i}-1\right]$;
• otherwise $\mathbf{bu}\left[\mathit{i}-1\right]-\mathbf{bl}\left[\mathit{i}-1\right]\ge 2.0\times \mathbf{rhobeg}$, for $\mathit{i}=1,2,\dots ,\mathbf{n}$.

7:     rhobeg – doubleInput

On entry: an initial lower bound on the value of the trust-region radius.

Suggested value: rhobeg should be about one tenth of the greatest expected overall change to a variable: the initial quadratic model will be constructed by taking steps from the initial x of length rhobeg along each coordinate direction.

Constraints:

• $\mathbf{rhobeg}>0.0$;
• $\mathbf{rhobeg}\ge \mathbf{rhoend}$.

8:     rhoend – doubleInput

On entry: a final lower bound on the value of the trust-region radius.

Suggested value: rhoend should indicate the absolute accuracy that is required in the final values of the variables.

Constraint: $\mathbf{rhoend}>0.0$.

9:     monfun – function, supplied by the userExternal Function

monfun may be used to monitor the optimization process. It is invoked every time a new trust-region radius is chosen.

If no monitoring is required, monfun may be specified as NULLFN.

The specification of monfun is:

void  monfun (Integer n, Integer nf, const double x[], double f, double rho, Nag_Comm *comm, Integer *inform)

1:     n – IntegerInput

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

2:     nf – IntegerInput

On entry: the cumulative number of calls made to objfun.

3:     x[n] – const doubleInput

On entry: the current best point.

4:     f – doubleInput

On entry: the value of objfun at x.

5:     rho – doubleInput

On entry: a lower bound on the current trust-region radius.

6:     comm – Nag_Comm *

Pointer to structure of type Nag_Comm; the following members are relevant to monfun.

user – double *

iuser – Integer *

p – Pointer 

The type Pointer will be void *. Before calling nag_opt_bounds_qa_no_deriv (e04jcc) you may allocate memory and initialize these pointers with various quantities for use by monfun when called from nag_opt_bounds_qa_no_deriv (e04jcc) (see Section 3.2.1.1 in the Essential Introduction).

7:     inform – Integer *Output

On exit: must be set to a value describing the action to be taken by the solver on return from monfun. Specifically, if the value is negative the solution of the current problem will terminate immediately; otherwise, computations will continue.

10:   maxcal – IntegerInput

On entry: the maximum permitted number of calls to objfun.

Constraint: $\mathbf{maxcal}\ge 1$.

11:   f – double *Output

On exit: the function value at the lowest point found (x).

12:   nf – Integer *Output

On exit: unless $\mathbf{fail}\mathbf{.}\mathbf{code}=$ NE_RESCUE_FAILED, NE_TOO_MANY_FEVALS, NE_TR_STEP_FAILED or NE_USER_STOP on exit, the total number of calls made to objfun.

13:   comm – Nag_Comm *Communication Structure

The NAG communication argument (see Section 3.2.1.1 in the Essential Introduction).

14:   fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

nag_opt_bounds_qa_no_deriv (e04jcc) returns with $\mathbf{fail}\mathbf{.}\mathbf{code}=$ NE_NOERROR if the final trust-region radius has reached its lower bound rhoend.

6  Error Indicators and Warnings

NE_ALLOC_FAIL

Dynamic memory allocation failed.

NE_BAD_PARAM

On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.

NE_BOUND

On entry, $\mathbf{rhobeg}=⟨\mathit{\text{value}}⟩$, $\mathbf{bl}\left[i-1\right]=⟨\mathit{\text{value}}⟩$, $\mathbf{bu}\left[i-1\right]=⟨\mathit{\text{value}}⟩$ and $i=⟨\mathit{\text{value}}⟩$.
Constraint: if $\mathbf{bl}\left[i-1\right]\ne \mathbf{bu}\left[i-1\right]$ in coordinate $i$, then $\mathbf{bu}\left[i-1\right]-\mathbf{bl}\left[i-1\right]\ge 2\times \mathbf{rhobeg}$.

NE_INT

On entry, $\mathbf{maxcal}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{maxcal}\ge 1$.

There were $n_r=⟨\mathit{\text{value}}⟩$ unequal bounds.
Constraint: $n_r\ge 2$.

There were $n_r=⟨\mathit{\text{value}}⟩$ unequal bounds and $\mathbf{npt}=⟨\mathit{\text{value}}⟩$ on entry.
Constraint: $n_r+2\le \mathbf{npt}\le \frac{\left(n_r+1\right)\times \left(n_r+2\right)}{2}$.

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.

NE_REAL

On entry, $\mathbf{rhobeg}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{rhobeg}>0.0$.

On entry, $\mathbf{rhoend}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{rhoend}>0.0$.

NE_REAL_2

On entry, $\mathbf{rhobeg}=⟨\mathit{\text{value}}⟩$ and $\mathbf{rhoend}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{rhoend}\le \mathbf{rhobeg}$.

NE_RESCUE_FAILED

A rescue procedure has been called in order to correct damage from rounding errors when computing an update to a quadratic approximation of $F$, but no further progess could be made. Check your specification of objfun and whether the function needs rescaling. Try a different initial x.

NE_TOO_MANY_FEVALS

The function evaluations limit was reached: objfun has been called maxcal times.

NE_TR_STEP_FAILED

The predicted reduction in a trust-region step was non-positive. Check your specification of objfun and whether the function needs rescaling. Try a different initial x.

NE_USER_STOP

User-supplied monitoring function requested termination.

User-supplied objective function requested termination.

7  Accuracy

8  Parallelism and Performance

9  Further Comments

10  Example

10.1  Program Text

Program Text (e04jcce.c)

10.2  Program Data

10.3  Program Results

Program Results (e04jcce.r)