IMSL C# Numerical Library

MinUncon Class

Finds the minimum point for a smooth univariate function using function and optionally first derivative evaluations.

For a list of all members of this type, see MinUncon Members.

System.Object
   Imsl.Math.MinUncon

public class MinUncon

Thread Safety

Public static (Shared in Visual Basic) members of this type are safe for multithreaded operations. Instance members are not guaranteed to be thread-safe.

Remarks

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 is a MinUncon.IFunction. Optional data include an initial estimate of the solution, and a positive number specified by the Bound property. Let x_0= Guess where Guess is specified by the Guess property 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 Step property. If Step 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 Accuracy property 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 is a MinUncon.IDerivative 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. The 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.

Requirements

Namespace: Imsl.Math

Assembly: ImslCS (in ImslCS.dll)

See Also

MinUncon Members | Imsl.Math Namespace | Function values only Example | Function values and derivatives Example