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20.2 Minimizers

Often it is useful to find the minimum value of a function rather than just the zeroes where it crosses the x-axis. fminbnd is designed for the simpler, but very common, case of a univariate function where the interval to search is bounded. For unbounded minimization of a function with potentially many variables use fminunc or fminsearch. The two functions use different internal algorithms and some knowledge of the objective function is required. For functions which can be differentiated, fminunc is appropriate. For functions with discontinuities, or for which a gradient search would fail, use fminsearch. See Optimization, for minimization with the presence of constraint functions. Note that searches can be made for maxima by simply inverting the objective function (Fto_max = -Fto_min).

x = fminbnd (fun, a, b)
x = fminbnd (fun, a, b, options)
[x, fval, info, output] = fminbnd (…)

Find a minimum point of a univariate function.

fun must be a function handle or name.

The starting interval is specified by a (left boundary) and b (right boundary). The endpoints must be finite.

options is a structure specifying additional parameters which control the algorithm. Currently, fminbnd recognizes these options: "Display", "FunValCheck", "MaxFunEvals", "MaxIter", "OutputFcn", "TolX".

"MaxFunEvals" proscribes the maximum number of function evaluations before optimization is halted. The default value is 500. The value must be a positive integer.

"MaxIter" proscribes the maximum number of algorithm iterations before optimization is halted. The default value is 500. The value must be a positive integer.

"TolX" specifies the termination tolerance for the solution x. The default is 1e-4.

For a description of the other options, see optimset. To initialize an options structure with default values for fminbnd use options = optimset ("fminbnd").

On exit, the function returns x, the approximate minimum point, and fval, the function evaluated x.

The third output info reports whether the algorithm succeeded and may take one of the following values:

• 1 The algorithm converged to a solution.
• 0 Iteration limit (either MaxIter or MaxFunEvals) exceeded.
• -1 The algorithm was terminated by a user OutputFcn.

Programming Notes: The search for a minimum is restricted to be in the finite interval bound by a and b. If you have only one initial point to begin searching from then you will need to use an unconstrained minimization algorithm such as fminunc or fminsearch. fminbnd internally uses a Golden Section search strategy.

fminunc (fcn, x0)
fminunc (fcn, x0, options)
[x, fval, info, output, grad, hess] = fminunc (fcn, …)

Solve an unconstrained optimization problem defined by the function fcn.

fcn should accept a vector (array) defining the unknown variables, and return the objective function value, optionally with gradient. fminunc attempts to determine a vector x such that fcn (x) is a local minimum.

x0 determines a starting guess. The shape of x0 is preserved in all calls to fcn, but otherwise is treated as a column vector.

options is a structure specifying additional parameters which control the algorithm. Currently, fminunc recognizes these options: "AutoScaling", "FinDiffType", "FunValCheck", "GradObj", "MaxFunEvals", "MaxIter", "OutputFcn", "TolFun", "TolX", "TypicalX".

If "AutoScaling" is "on", the variables will be automatically scaled according to the column norms of the (estimated) Jacobian. As a result, "TolFun" becomes scaling-independent. By default, this option is "off" because it may sometimes deliver unexpected (though mathematically correct) results.

If "GradObj" is "on", it specifies that fcn—when called with two output arguments—also returns the Jacobian matrix of partial first derivatives at the requested point.

"MaxFunEvals" proscribes the maximum number of function evaluations before optimization is halted. The default value is 100 * number_of_variables, i.e., 100 * length (x0). The value must be a positive integer.

"MaxIter" proscribes the maximum number of algorithm iterations before optimization is halted. The default value is 400. The value must be a positive integer.

"TolX" specifies the termination tolerance for the unknown variables x, while "TolFun" is a tolerance for the objective function value fval. The default is 1e-6 for both options.

For a description of the other options, see optimset.

On return, x is the location of the minimum and fval contains the value of the objective function at x.

info may be one of the following values:

1

Converged to a solution point. Relative gradient error is less than specified by TolFun.

2

Last relative step size was less than TolX.

3

Last relative change in function value was less than TolFun.

0

Iteration limit exceeded—either maximum number of algorithm iterations MaxIter or maximum number of function evaluations MaxFunEvals.

-1

Algorithm terminated by OutputFcn.

-3

The trust region radius became excessively small.

Optionally, fminunc can return a structure with convergence statistics (output), the output gradient (grad) at the solution x, and approximate Hessian (hess) at the solution x.

Application Notes: If the objective function is a single nonlinear equation of one variable then using fminbnd is usually a better choice.

The algorithm used by fminunc is a gradient search which depends on the objective function being differentiable. If the function has discontinuities it may be better to use a derivative-free algorithm such as fminsearch.

x = fminsearch (fun, x0)
x = fminsearch (fun, x0, options)
x = fminsearch (problem)
[x, fval, exitflag, output] = fminsearch (…)

Find a value of x which minimizes the multi-variable function fun.

The search begins at the point x0 and iterates using the Nelder & Mead Simplex algorithm (a derivative-free method). This algorithm is better-suited to functions which have discontinuities or for which a gradient-based search such as fminunc fails.

Options for the search are provided in the parameter options using the function optimset. Currently, fminsearch accepts the options: "Display", "FunValCheck","MaxFunEvals", "MaxIter", "OutputFcn", "TolFun", "TolX".

"MaxFunEvals" proscribes the maximum number of function evaluations before optimization is halted. The default value is 200 * number_of_variables, i.e., 200 * length (x0). The value must be a positive integer.

"MaxIter" proscribes the maximum number of algorithm iterations before optimization is halted. The default value is 200 * number_of_variables, i.e., 200 * length (x0). The value must be a positive integer.

For a description of the other options, see optimset. To initialize an options structure with default values for fminsearch use options = optimset ("fminsearch").

fminsearch may also be called with a single structure argument with the following fields:

objective

The objective function.

x0

The initial point.

solver

Must be set to "fminsearch".

options

A structure returned from optimset or an empty matrix to indicate that defaults should be used.

The field options is optional. All others are required.

On exit, the function returns x, the minimum point, and fval, the function value at the minimum.

The third output exitflag reports whether the algorithm succeeded and may take one of the following values:

1

if the algorithm converged (size of the simplex is smaller than TolX AND the step in function value between iterations is smaller than TolFun).

0

if the maximum number of iterations or the maximum number of function evaluations are exceeded.

-1

if the iteration is stopped by the "OutputFcn".

The fourth output is a structure output containing runtime about the algorithm. Fields in the structure are funcCount containing the number of function calls to fun, iterations containing the number of iteration steps, algorithm with the name of the search algorithm (always: "Nelder-Mead simplex direct search"), and message with the exit message.

Example:

fminsearch (@(x) (x(1)-5).^2+(x(2)-8).^4, [0;0])

Note: If you need to find the minimum of a single variable function it is probably better to use fminbnd.

The function humps is a useful function for testing zero and extrema finding functions.

y = humps (x)
[x, y] = humps (x)

Evaluate a function with multiple minima, maxima, and zero crossings.

The output y is the evaluation of the rational function:

1200*x^4 - 2880*x^3 + 2036*x^2 - 348*x - 88
y = - ---------------------------------------------
200*x^4 - 480*x^3 + 406*x^2 - 138*x + 17

x may be a scalar, vector or array. If x is omitted, the default range [0:0.05:1] is used.

When called with two output arguments, [x, y], x will contain the input values, and y will contain the output from humps.

Programming Notes: humps has two local maxima located near x = 0.300 and 0.893, a local minimum near x = 0.637, and zeros near x = -0.132 and 1.300. humps is a useful function for testing algorithms which find zeros or local minima and maxima.

Try demo humps to see a plot of the humps function.