fminbnd - Find minimum of single-variable function on fixed interval

Equation

Finds a minimum for a problem specified by

x, x₁, and x₂ are scalars and f(x) is a function that returns a scalar.

Syntax

x = fminbnd(fun,x1,x2) x = fminbnd(fun,x1,x2,options) x = fminbnd(problem) [x,fval] = fminbnd(...) [x,fval,exitflag] = fminbnd(...) [x,fval,exitflag,output] = fminbnd(...)

Description

fminbnd attempts to find a minimum of a function of one variable within a fixed interval.

Note Passing Extra Parameters explains how to pass extra parameters to the objective function, if necessary.

x = fminbnd(fun,x1,x2) returns a value x that is a local minimizer of the scalar valued function that is described in fun in the interval x1 < x < x2. fun is a function handle for either an M-file function or an anonymous function.

x = fminbnd(fun,x1,x2,options) minimizes with the optimization options specified in the structure options. Use optimset to set these options.

x = fminbnd(problem) finds the minimum for problem, where problem is a structure described in Input Arguments.

Create the structure problem by exporting a problem from Optimization Tool, as described in Exporting to the MATLAB Workspace.

[x,fval] = fminbnd(...) returns the value of the objective function computed in fun at the solution x.

[x,fval,exitflag] = fminbnd(...) returns a value exitflag that describes the exit condition of fminbnd.

[x,fval,exitflag,output] = fminbnd(...) returns a structure output that contains information about the optimization.

Input Arguments

Function Arguments contains general descriptions of arguments passed into fminbnd. This section provides function-specific details for fun, options, and problem:

`fun`	The function to be minimized. `fun` is a function handle for a function that accepts a scalar `x` and returns a scalar `f`, the objective function evaluated at `x`. The function `fun` can be specified as a function handle for an M-file function x = fminbnd(@myfun,x1,x2) where `myfun` is a MATLAB function such as function f = myfun(x) f = ... % Compute function value at x. `fun` can also be a function handle for an anonymous function. x = fminbnd(@(x)sin(x^2),x1,x2);
`options`	Options provides the function-specific details for the `options` values.
`problem`	`f`	Objective function
	`x1`	Left endpoint
	`x2`	Right endpoint
	`solver`	`'fminbnd'`
	`options`	Options structure created with `optimset`

Output Arguments

Function Arguments contains general descriptions of arguments returned by fminbnd. This section provides function-specific details for exitflag and output:

`exitflag`	Integer identifying the reason the algorithm terminated. The following lists the values of `exitflag` and the corresponding reasons the algorithm terminated.
	`1`	Function converged to a solution `x`.
	`0`	Number of iterations exceeded `options.MaxIter` or number of function evaluations exceeded `options.FunEvals`.
	`-1`	Algorithm was terminated by the output function.
	`-2`	The bounds are inconsistent.
`output`	Structure containing information about the optimization. The fields of the structure are
	`iterations`	Number of iterations taken
	`funcCount`	Number of function evaluations
	`algorithm`	Optimization algorithm used
	`message`	Exit message

Options

Optimization options used by fminbnd. You can use optimset to set or change the values of these fields in the options structure options. See Optimization Options for detailed information.

`Display`	Level of display. `'off'` displays no output; `'iter'` displays output at each iteration; `'final'` displays just the final output; `'notify'` (default) displays output only if the function does not converge.
`FunValCheck`	Check whether objective function values are valid. `'on'` displays an error when the objective function returns a value that is `complex`, `Inf`, or `NaN`. The default `'off'` displays no error.
`MaxFunEvals`	Maximum number of function evaluations allowed, a positive integer. The default is `500`.
`MaxIter`	Maximum number of iterations allowed, a positive integer. The default is `500`.
`OutputFcn`	Specify one or more user-defined functions that an optimization function calls at each iteration, either as a function handle or as a cell array of function handles. The default is none (`[]`). See Output Function.
`PlotFcns`	Plots various measures of progress while the algorithm executes, select from predefined plots or write your own. Pass a function handle or a cell array of function handles. The default is none (`[]`). Specifying `@optimplotx` plots the current point; `@optimplotfunccount` plots the function count; `@optimplotfval` plots the function value.
`TolX`	Termination tolerance on `x`, a positive scalar. The default is `1e-4`.

Examples

A minimum of sin(x) occurs at

x = fminbnd(@sin,0,2*pi)
x = 
     4.7124

The value of the function at the minimum is

y = sin(x)
y = 
    -1.0000

To find the minimum of the function

f(x) = (x – 3)² – 1,

on the interval (0,5), first write an M-file.

function f = myfun(x)
f = (x-3)^2 - 1;

Next, call an optimization routine.

x = fminbnd(@myfun,0,5)

This generates the solution

x =
     3

The value at the minimum is

y = f(x)

y =
    -1

If fun is parameterized, you can use anonymous functions to capture the problem-dependent parameters. For example, suppose you want to minimize the objective function myfun defined by the following M-file function.

function f = myfun(x,a)
f = (x - a)^2;

Note that myfun has an extra parameter a, so you cannot pass it directly to fminbind. To optimize for a specific value of a, such as a = 1.5.

Assign the value to a.
```
a = 1.5; % define parameter first
```
Call fminbnd with a one-argument anonymous function that captures that value of a and calls myfun with two arguments:
```
x = fminbnd(@(x) myfun(x,a),0,1)
```

Algorithm

fminbnd is an M-file. The algorithm is based on golden section search and parabolic interpolation. Unless the left endpoint x₁ is very close to the right endpoint x₂, fminbnd never evaluates fun at the endpoints, so fun need only be defined for x in the interval x₁ < x < x₂. If the minimum actually occurs at x₁ or x₂, fminbnd returns an interior point at a distance of no more than 2*TolX from x₁ or x₂, where TolX is the termination tolerance. See [1] or [2] for details about the algorithm.

Limitations

The function to be minimized must be continuous. fminbnd might only give local solutions.

fminbnd often exhibits slow convergence when the solution is on a boundary of the interval. In such a case, fmincon often gives faster and more accurate solutions.

fminbnd only handles real variables.

References

[1] Forsythe, G.E., M.A. Malcolm, and C.B. Moler, Computer Methods for Mathematical Computations, Prentice Hall, 1976.

[2] Brent, Richard. P., Algorithms for Minimization without Derivatives, Prentice-Hall, Englewood Cliffs, New Jersey, 1973.