e04ly is an easy-to-use modified-Newton algorithm for finding a minimum of a function,

F (x_{1}, x_{2}, \dots, x_{n})

subject to fixed upper and lower bounds on the independent variables,

x_{1}, x_{2}, \dots, x_{n}

when first and second derivatives of

F

are available. It is intended for functions which are continuous and which have continuous first and second derivatives (although it will usually work even if the derivatives have occasional discontinuities).

Syntax

C#
public static void e04ly( int n, int ibound, E04..::..E04LY_FUNCT2 funct2, E04..::..E04LY_HESS2 hess2, double[] bl, double[] bu, double[] x, out double f, double[] g, out int ifail )

Visual Basic
Public Shared Sub e04ly ( _ n As Integer, _ ibound As Integer, _ funct2 As E04..::..E04LY_FUNCT2, _ hess2 As E04..::..E04LY_HESS2, _ bl As Double(), _ bu As Double(), _ x As Double(), _ <OutAttribute> ByRef f As Double, _ g As Double(), _ <OutAttribute> ByRef ifail As Integer _ )

Visual Basic

Public Shared Sub e04ly ( _
	n As Integer, _
	ibound As Integer, _
	funct2 As E04..::..E04LY_FUNCT2, _
	hess2 As E04..::..E04LY_HESS2, _
	bl As Double(), _
	bu As Double(), _
	x As Double(), _
	<OutAttribute> ByRef f As Double, _
	g As Double(), _
	<OutAttribute> ByRef ifail As Integer _
)

Visual C++
public: static void e04ly( int n, int ibound, E04..::..E04LY_FUNCT2^ funct2, E04..::..E04LY_HESS2^ hess2, array<double>^ bl, array<double>^ bu, array<double>^ x, [OutAttribute] double% f, array<double>^ g, [OutAttribute] int% ifail )

Visual C++

public:
static void e04ly(
	int n, 
	int ibound, 
	E04..::..E04LY_FUNCT2^ funct2, 
	E04..::..E04LY_HESS2^ hess2, 
	array<double>^ bl, 
	array<double>^ bu, 
	array<double>^ x, 
	[OutAttribute] double% f, 
	array<double>^ g, 
	[OutAttribute] int% ifail
)

F#
static member e04ly : n : int * ibound : int * funct2 : E04..::..E04LY_FUNCT2 * hess2 : E04..::..E04LY_HESS2 * bl : float[] * bu : float[] * x : float[] * f : float byref * g : float[] * ifail : int byref -> unit

static member e04ly : 
        n : int * 
        ibound : int * 
        funct2 : E04..::..E04LY_FUNCT2 * 
        hess2 : E04..::..E04LY_HESS2 * 
        bl : float[] * 
        bu : float[] * 
        x : float[] * 
        f : float byref * 
        g : float[] * 
        ifail : int byref -> unit

Parameters

n: Type: System..::..Int32
On entry: the number $n$ of independent variables.

Constraint: $n \geq 1$ .

ibound

Type: System..::..Int32

On entry: indicates whether the facility for dealing with bounds of special forms is to be used. It must be set to one of the following values:

$ibound = 0$: If you are supplying all the $l_{j}$ and $u_{j}$ individually.
$ibound = 1$: If there are no bounds on any $x_{j}$ .
$ibound = 2$: If all the bounds are of the form $0 \leq x_{j}$ .
$ibound = 3$: If $l_{1} = l_{2} = \dots = l_{n}$ and $u_{1} = u_{2} = \dots = u_{n}$ .

Constraint:

0 \leq ibound \leq 3

funct2: Type: NagLibrary..::..E04..::..E04LY_FUNCT2
You must supply this method to calculate the values of the function $F (x)$ and its first derivatives $\frac{\partial F}{\partial x_{j}}$ at any point $x$ . It should be tested separately before being used in conjunction with e04ly (see the E04 class).
A delegate of type E04LY_FUNCT2.

hess2: Type: NagLibrary..::..E04..::..E04LY_HESS2
You must supply this method to evaluate the elements $H_{i j} = \frac{\partial^{2} F}{\partial x_{i} \partial x_{j}}$ of the matrix of second derivatives of $F (x)$ at any point $x$ . It should be tested separately before being used in conjunction with e04ly (see the E04 class).
A delegate of type E04LY_HESS2.

bl: Type: array<System..::..Double>[]()[][]
An array of size [n]
On entry: the lower bounds $l_{j}$ .
If ibound is set to $0$ , $bl [j - 1]$ must be set to $l_{j}$ , for $j = 1, 2, \dots, n$ . (If a lower bound is not specified for any $x_{j}$ , the corresponding $bl [j - 1]$ should be set to $- 10^{6}$ .)

If ibound is set to $3$ , you must set $bl [0]$ to $l_{1}$ ; e04ly will then set the remaining elements of bl equal to $bl [0]$ .

On exit: the lower bounds actually used by e04ly.

bu: Type: array<System..::..Double>[]()[][]
An array of size [n]
On entry: the upper bounds $u_{j}$ .
If ibound is set to $0$ , $bu [j - 1]$ must be set to $u_{j}$ , for $j = 1, 2, \dots, n$ . (If an upper bound is not specified for any $x_{j}$ the corresponding $bu [j - 1]$ should be set to $10^{6}$ .)

If ibound is set to $3$ , you must set $bu [0]$ to $u_{1}$ ; e04ly will then set the remaining elements of bu equal to $bu [0]$ .

On exit: the upper bounds actually used by e04ly.

x: Type: array<System..::..Double>[]()[][]
An array of size [n]
On entry: $x [j - 1]$ must be set to a guess at the $j$ th component of the position of the minimum, for $j = 1, 2, \dots, n$ . The method checks the gradient and the Hessian matrix at the starting point, and is more likely to detect any error in your programming if the initial $x [j - 1]$ are nonzero and mutually distinct.
On exit: the lowest point found during the calculations. Thus, if $ifail = 0$ on exit, $x [j - 1]$ is the $j$ th component of the position of the minimum.

f: Type: System..::..Double%
On exit: the value of $F (x)$ corresponding to the final point stored in x.

g: Type: array<System..::..Double>[]()[][]
An array of size [n]
On exit: the value of $\frac{\partial F}{\partial x_{j}}$ corresponding to the final point stored in x, for $j = 1, 2, \dots, n$ ; the value of $g [j - 1]$ for variables not on a bound should normally be close to zero.

ifail: Type: System..::..Int32%
On exit: $ifail = 0$ unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

Description

e04ly is applicable to problems of the form:

Minimize F (x_{1}, x_{2}, \dots, x_{n}) subject to l_{j} \leq x_{j} \leq u_{j}, j = 1, 2, \dots, n

when first and second derivatives of

F (x)

are available.

Special provision is made for problems which actually have no bounds on the

x_{j}

, problems which have only non-negativity bounds and problems in which

l_{1} = l_{2} = \dots = l_{n}

and

u_{1} = u_{2} = \dots = u_{n}

. You must supply a method to calculate the values of

F (x)

and its first derivatives at any point

x

and a method to calculate the second derivatives.

From a starting point you supplied there is generated, on the basis of estimates of the curvature of

F (x)

, a sequence of feasible points which is intended to converge to a local minimum of the constrained function.

References

Gill P E and Murray W (1976) Minimization subject to bounds on the variables NPL Report NAC 72 National Physical Laboratory

Error Indicators and Warnings

Note: e04ly may return useful information for one or more of the following detected errors or warnings.

Errors or warnings detected by the method:

$ifail = 1$

On entry,	$n < 1$ ,
or	$ibound < 0$ ,
or	$ibound > 3$ ,
or	$ibound = 0$ and $bl [j - 1] > bu [j - 1]$ for some $j$ ,
or	$ibound = 3$ and $bl [0] > bu [0]$ ,
or	$liw < n + 2$ ,
or	$lw < \max (10, n \times (n + 7))$ .

$ifail = 2$: There have been $50 \times n$ function evaluations, yet the algorithm does not seem to be converging. The calculations can be restarted from the final point held in x. The error may also indicate that $F (x)$ has no minimum.

$ifail = 3$: The conditions for a minimum have not all been met but a lower point could not be found and the algorithm has failed.

$ifail = 4$: Not used. (This value of the parameter is included so as to make the significance of $ifail = 5$ etc. consistent in the easy-to-use methods.)

$ifail = 5$
$ifail = 6$
$ifail = 7$
$ifail = 8$: There is some doubt about whether the point $x$ found by e04ly is a minimum. The degree of confidence in the result decreases as ifail increases. Thus, when $ifail = 5$ it is probable that the final $x$ gives a good estimate of the position of a minimum, but when $ifail = 8$ it is very unlikely that the method has found a minimum.

$ifail = 9$: In the search for a minimum, the modulus of one of the variables has become very large $(\sim 10^{6})$ . This indicates that there is a mistake in user-supplied delegates funct2 or hess2, that your problem has no finite solution, or that the problem needs rescaling (see [Further Comments]).

$ifail = 10$: It is very likely that you have made an error in forming the gradient.

$ifail = 11$: It is very likely that you have made an error in forming the second derivatives.

$ifail = -9000$: An error occured, see message report.
$ifail = -8000$: Negative dimension for array $〈value〉$
$ifail = -6000$: Invalid Parameters $〈value〉$

If you are dissatisfied with the result (e.g., because

ifail = 5

6

7

8

), it is worth restarting the calculations from a different starting point (not the point at which the failure occurred) in order to avoid the region which caused the failure.

Accuracy

When a successful exit is made then, for a computer with a mantissa of

t

decimals, one would expect to get about

t / 2 - 1

decimals accuracy in

x

, and about

t - 1

decimals accuracy in

F

, provided the problem is reasonably well scaled.

Parallelism and Performance

None.

Further Comments

The number of iterations required depends on the number of variables, the behaviour of

F (x)

and the distance of the starting point from the solution. The number of operations performed in an iteration of e04ly is roughly proportional to

n^{3} + O (n^{2})

. In addition, each iteration makes one call of hess2 and at least one call of funct2. So, unless

F (x)

, the gradient vector and the matrix of second derivatives can be evaluated very quickly, the run time will be dominated by the time spent in user-supplied delegates funct2 and hess2.

Ideally the problem should be scaled so that at the solution the value of

F (x)

and the corresponding values of

x_{1}, x_{2}, \dots x_{n}

are each in the range

(- 1, + 1)

, and so that at points a unit distance away from the solution,

F

is approximately a unit value greater than at the minimum. It is unlikely that you will be able to follow these recommendations very closely, but it is worth trying (by guesswork), as sensible scaling will reduce the difficulty of the minimization problem, so that e04ly will take less computer time.

Example

A program to minimize

F = {(x_{1} + 10 x_{2})}^{2} + 5 {(x_{3} - x_{4})}^{2} + {(x_{2} - 2 x_{3})}^{4} + 10 {(x_{1} - x_{4})}^{4}

subject to

\begin{aligned} 1 & \leq & x_{1} & \leq & 3 \\ - 2 & \leq & x_{2} & \leq & 0 \\ 1 & \leq & x_{4} & \leq & 3 . \end{aligned}

starting from the initial guess

(3, - 1, 0, 1)

. (In practice, it is worth trying to make user-supplied delegates funct2 and hess2 as efficient as possible. This has not been done in the example program for reasons of clarity.)

Example program (C#): e04lye.cs

Example program results: e04lye.r