public class MinUncon extends Object implements Serializable, Cloneable
MinUncon uses two separate algorithms to compute the
minimum depending on what the user supplies as the function
f defines the function whose minimum is to be found
MinUncon uses a safeguarded quadratic interpolation method to
find a minimum point of a univariate function. Both the code and the
underlying algorithm are based on the routine
ZXLSF written by
M.J.D. Powell at the University of Cambridge.
MinUncon finds the least value of a univariate function,
Optional data include an initial estimate of the solution, and a positive
bound, specified by the
xguess is specified
setGuess method and b = bound, then
x is restricted to the interval
. Usually, the algorithm begins the
search by moving from to ,
where s = step.
step is set by the
setStep method. If
setStep is not called then
step is set to 0.1.
may be positive or negative. The first two function evaluations indicate the
direction to the minimum point, and the search strides out along this
direction until a bracket on a minimum point is found or until
x reaches one of the bounds .
During this stage, the step length increases by a factor of between two and
nine per function evaluation; the factor depends on the position of the
minimum point that is predicted by quadratic interpolation of the three most
recent function values.
When an interval containing a solution has been found, we will have three points, , , and , with and and . There are three main ingredients in the technique for choosing the new x from these three points. They are (i) the estimate of the minimum point that is given by quadratic interpolation of the three function values, (ii) a tolerance parameter , that depends on the closeness of f to a quadratic, and (iii) whether is near the center of the range between and or is relatively close to an end of this range. In outline, the new value of x is as near as possible to the predicted minimum point, subject to being at least from , and subject to being in the longer interval between and or and when is particularly close to or . There is some elaboration, however, when the distance between these points is close to the required accuracy; when the distance is close to the machine precision; or when is relatively large.
The algorithm is intended to provide fast convergence when f has a positive and continuous second derivative at the minimum and to avoid gross inefficiencies in pathological cases, such as
The algorithm can make large automatically
in the pathological cases. In this case, it is usual for a new value of
x to be at the midpoint of the longer interval that is
adjacent to the least calculated function value. The midpoint strategy is used
frequently when changes to f are dominated by
computer rounding errors, which will
almost certainly happen if the user requests an accuracy that is less than
the square root of the machine precision. In such cases, the routine claims
to have achieved the required accuracy if it knows that there is a local
minimum point within distance of
x, where , specified
setAccuracy method even though the rounding errors in
f may cause the existence of other local minimum points
nearby. This difficulty is inevitable in minimization routines that use only
function values, so high precision arithmetic is recommended.
MinUncon uses a descent method with either the secant method or
cubic interpolation to find a minimum point of a univariate function. It
starts with an initial guess and two endpoints. If any of the three points
is a local minimum point and has least function value, the routine
terminates with a solution. Otherwise, the point with least function value
will be used as the starting point.
From the starting point, say , the function value , the derivative value , and a new point defined by are computed. The function , and the derivative are then evaluated. If either or has the opposite sign of , then there exists a minimum point between and ; and an initial interval is obtained. Otherwise, since is kept as the point that has lowest function value, an interchange between and is performed. The secant method is then used to get a new point
Let and repeat this process until an interval containing a minimum is found or one of the convergence criteria is satisfied. The convergence criteria are as follows: Criterion 1:
where , is a relative error tolerance and is a gradient tolerance.
When convergence is not achieved, a cubic interpolation is performed to obtain a new point. Function and derivative are then evaluated at that point; and accordingly, a smaller interval that contains a minimum point is chosen. A safeguarded method is used to ensure that the interval reduces by at least a fraction of the previous interval. Another cubic interpolation is then performed, and this procedure is repeated until one of the stopping criteria is met.
|Modifier and Type||Class and Description|
Public interface for the user supplied function to the
Public interface for the user supplied function to the
|Constructor and Description|
Unconstrained minimum constructor for a smooth function of a single variable of type
|Modifier and Type||Method and Description|
Return the minimum of a smooth function of a single variable of type
Set the required absolute accuracy in the final value returned by member function
Set the amount by which X may be changed from its initial value, xguess.
Set the derivative tolerance used by member function
Set the initial guess of the minimum point of the input function.
Set the stepsize to use when changing x.
public double computeMin(MinUncon.Function F)
doubleusing function values only or using function values and derivatives.
F- defines the function whose minimum is to be found. If
Derivativethen derivatives are used. Otherwise, an attempt to find the minimum is made using function values only.
doublescalar value containing the minimum of the input function
public void setAccuracy(double xacc)
computeMin. If this member function is not called, the required accuracy is set to 1.0e-8.
doublescalar value specifying the required absolute accuracy in the final value returned by member function
public void setBound(double bound)
doublescalar value specifying the amount by which X may be changed from its initial value. In other words, X is restricted to the interval [
public void setDerivtol(double gtol)
computeMinto decide if the current point is a local minimum. This is the second stopping criterion. x is returned as a solution when G(x) is less than or equal to
gtolshould be nonnegative, otherwise zero will be used. If this member function is not called, the derivative tolerance is set to 1.0e-8.
doublescalar value specifying the derivative tolerance used by member function
public void setGuess(double xguess)
doublescalar value specifying the initial guess of the minimum point of the input function
public void setStep(double step)
doublescalar value specifying the order of magnitude estimate of the required change in x when stepping towards the minimum
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Built October 13 2015.