e04kz is an easy-to-use modified Newton algorithm for finding a minimum of a function , subject to fixed upper and lower bounds on the independent variables , when first derivatives of 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).
Public Shared Sub e04kz ( _ n As Integer, _ ibound As Integer, _ funct2 As E04..::..E04KZ_FUNCT2, _ bl As Double(), _ bu As Double(), _ x As Double(), _ <OutAttribute> ByRef f As Double, _ g As Double(), _ <OutAttribute> ByRef ifail As Integer _ )
public: static void e04kz( int n, int ibound, E04..::..E04KZ_FUNCT2^ funct2, array<double>^ bl, array<double>^ bu, array<double>^ x, [OutAttribute] double% f, array<double>^ g, [OutAttribute] int% ifail )
- Type: System..::..Int32On entry: the number of independent variables.Constraint: .
- Type: System..::..Int32On 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:
- If you are supplying all the and individually.
- If there are no bounds on any .
- If all the bounds are of the form .
- If and .
- Type: NagLibrary..::..E04..::..E04KZ_FUNCT2You must supply this method to calculate the values of the function and its first derivatives at any point . It should be tested separately before being used in conjunction with e04kz (see E04 class).
A delegate of type E04KZ_FUNCT2.
- Type: array<System..::..Double>()On entry: the lower bounds .If ibound is set to , you must set to , for . (If a lower bound is not specified for a particular , the corresponding should be set to .)On exit: the lower bounds actually used by e04kz.
- Type: array<System..::..Double>()On entry: the upper bounds .If ibound is set to , you must set to , for . (If an upper bound is not specified for a particular , the corresponding should be set to .)On exit: the upper bounds actually used by e04kz.
- Type: array<System..::..Double>()On entry: must be set to a guess at the th component of the position of the minimum, for . The method checks the gradient at the starting point, and is more likely to detect any error in your programming if the initial are nonzero and mutually distinct.On exit: the lowest point found during the calculations of the position of the minimum.
- Type: array<System..::..Double>()On exit: the value of corresponding to the final point stored in x, for ; the value of for variables not on a bound should normally be close to zero.
e04kz is applicable to problems of the form:
when first derivatives are known.
Special provision is made for problems which actually have no bounds on the , problems which have only non-negativity bounds, and problems in which and . You must supply a method to calculate the values of and its first derivatives at any point .
From a starting point you supplied there is generated, on the basis of estimates of the gradient of the curvature of , a sequence of feasible points which is intended to converge to a local minimum of the constrained function.
Gill P E and Murray W (1976) Minimization subject to bounds on the variables NPL Report NAC 72 National Physical Laboratory
Note: e04kz may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the method:
On entry, , or , or , or and for some , or and , or , or .
- There has been a large number of 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 has no minimum.
- The conditions for a minimum have not all been met but a lower point could not be found and the algorithm has failed.
- Not used. (This value of the parameter is included to make the significance of etc. consistent in the easy-to-use methods.)
- There is some doubt about whether the point found by e04kz is a minimum. The degree of confidence in the result decreases as ifail increases. Thus, when it is probable that the final gives a good estimate of the position of a minimum, but when it is very unlikely that the method has found a minimum.
- It is very likely that you have made an error in forming the gradient.
If you are dissatisfied with the result (e.g., because , , or ), 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. If persistent trouble occurs and it is possible to calculate second derivatives it may be advisable to change to a method which uses second derivatives (see the E04 class).
When a successful exit is made then, for a computer with a mantissa of decimals, one would expect to get about decimals accuracy in and about decimals accuracy in , provided the problem is reasonably well scaled.
The number of iterations required depends on the number of variables, the behaviour of and the distance of the starting point from the solution. The number of operations performed in an iteration of e04kz is roughly proportional to . In addition, each iteration makes at least calls of funct2 where is the number of variables not fixed on bounds. So unless and the gradient vector can be evaluated very quickly, the run time will be dominated by the time spent in funct2.
Ideally the problem should be scaled so that at the solution the value of and the corresponding values of are in the range , and so that at points a unit distance away from the solution, 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 e04kz will take less computer time.
A program to minimize
starting from the initial guess .
In practice, it is worth trying to make funct2 as efficient as possible. This has not been done in the example program for reasons of clarity.