com.imsl.math

Class MinUncon

java.lang.Object

com.imsl.math.MinUncon

All Implemented Interfaces:

Serializable, Cloneable

public class MinUncon
extends Object
implements Serializable, Cloneable

Unconstrained minimization.

MinUncon uses two separate algorithms to compute the minimum depending on what the user supplies as the function f.

If 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, f, where f implements MinUnconFunction f. Optional data include an initial estimate of the solution, and a positive number bound, specified by the setBound method. Let \(x_0= xguess\) where xguess is specified by the setGuess method and b = bound, then x is restricted to the interval \([x_0 - b, x_0 + b]\). Usually, the algorithm begins the search by moving from \(x_0\) to \(x = x_0 + s\), where s = step. step is set by the setStep method. If setStep is not called then step is set to 0.1. step 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 \(x_0 \pm b\). 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, \(x_1\), \(x_2\), and \(x_3\), with \(x_1 \lt x_2\lt x_3\) and \(f (x_2)\le f (x_1)\) and \(f (x_2) \le f (x_3)\). 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 \(\varepsilon\), that depends on the closeness of f to a quadratic, and (iii) whether \(x_2\) is near the center of the range between \(x_1\) and \(x_3\) 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 \(\varepsilon\) from \(x_2\), and subject to being in the longer interval between \( x_1\) and \(x_2\) or \(x_2\) and \(x_3\) when \( x_2\) is particularly close to \(x_1\) or \(x_3\). 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 \(\varepsilon\) 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

$$f\left( x \right) = x + 1.001\left| x \right|$$

The algorithm can make \(\varepsilon\) 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 \(\delta\) of x, where \(\delta = xacc\), specified by the 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.

If f implements MinUnconDerivative then 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 \(x_c\), the function value \(f_c = f (x_c)\), the derivative value \(g_c = g(x_c)\), and a new point \(x_n\) defined by \(x_n = x_c - g_c\) are computed. The function \(f_n = f(x_n)\), and the derivative \(g_n = g(x_n)\) are then evaluated. If either \(f_n \ge f_c\) or \(g_n\) has the opposite sign of \(g_c\), then there exists a minimum point between \(x_c\) and \(x_n\); and an initial interval is obtained. Otherwise, since \(x_c\) is kept as the point that has lowest function value, an interchange between \(x_n\) and \(x_c\) is performed. The secant method is then used to get a new point

$$x_s = x_c - g_c ({{g_n - g_c } \over {x_n - x_c }})$$

Let \(x_n \leftarrow x_s\) 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: $$\left| {x_c - x_n } \right| \le \;\varepsilon _c$$

Criterion 2:

$$\left| {g_c } \right| \le \;\varepsilon _g$$

where \(\varepsilon _c = \max \left\{ {1.0,\left| {x_c } \right|} \right\}\varepsilon\), \(\varepsilon\) is a relative error tolerance and \(\varepsilon_c\) 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.

See Also:

Function values only Example, Function values and derivatives Example, Serialized Form

Nested Class Summary

Nested Classes

Modifier and Type	Class and Description
static interface	MinUncon.Derivative Public interface for the user supplied function to the MinUncon object.
static interface	MinUncon.Function Public interface for the user supplied function to the MinUncon object.

Constructor Summary

Constructors

Constructor and Description
MinUncon() Unconstrained minimum constructor for a smooth function of a single variable of type double.

Method Summary

Modifier and Type	Method and Description
double	computeMin(MinUncon.Function F) Return the minimum of a smooth function of a single variable of type double using function values only or using function values and derivatives.
void	setAccuracy(double xacc) Set the required absolute accuracy in the final value returned by member function computeMin.
void	setBound(double bound) Set the amount by which X may be changed from its initial value, xguess.
void	setDerivtol(double gtol) Set the derivative tolerance used by member function computeMin to decide if the current point is a local minimum.
void	setGuess(double xguess) Set the initial guess of the minimum point of the input function.
void	setStep(double step) Set the stepsize to use when changing x.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Detail

MinUncon

public MinUncon()

Unconstrained minimum constructor for a smooth function of a single variable of type double.

Method Detail

computeMin

public double computeMin(MinUncon.Function F)

Return the minimum of a smooth function of a single variable of type double using function values only or using function values and derivatives.

Parameters:

F - defines the function whose minimum is to be found. If F implements Derivative then derivatives are used. Otherwise, an attempt to find the minimum is made using function values only.

Returns:

a double scalar value containing the minimum of the input function

setGuess

public void setGuess(double xguess)

Set the initial guess of the minimum point of the input function. If this member function is not called, an initial guess of 0.0 is used.

Parameters:

xguess - a double scalar value specifying the initial guess of the minimum point of the input function

setStep

public void setStep(double step)

Set the stepsize to use when changing x. If this member function is not called, step is set to 0.1.

Parameters:

step - a double scalar value specifying the order of magnitude estimate of the required change in x when stepping towards the minimum

setBound

public void setBound(double bound)

Set the amount by which X may be changed from its initial value, xguess. If this member function is not called, bound is set to 100.

Parameters:

bound - a double scalar value specifying the amount by which X may be changed from its initial value. In other words, X is restricted to the interval [xguess-bound, xguess+bound].

setAccuracy

public void setAccuracy(double xacc)

Set the required absolute accuracy in the final value returned by member function computeMin. If this member function is not called, the required accuracy is set to 1.0e-8.

Parameters:

xacc - a doublescalar value specifying the required absolute accuracy in the final value returned by member function computeMin.

setDerivtol

public void setDerivtol(double gtol)

Set the derivative tolerance used by member function computeMin to 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 gtol. gtol should be nonnegative, otherwise zero will be used. If this member function is not called, the derivative tolerance is set to 1.0e-8.

Parameters:

gtol - a doublescalar value specifying the derivative tolerance used by member function computeMin.