fminbnd (Optimization Toolbox)

Find the minimum of a function of one variable on a fixed interval

where 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(fun,x1,x2,options,P1,P2,...)
[x,fval] = fminbnd(...)
[x,fval,exitflag] = fminbnd(...)
[x,fval,exitflag,output] = fminbnd(...)

Description

fminbnd finds the minimum of a function of one variable within a fixed interval.

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.

x = fminbnd(fun,x1,x2,options) minimizes with the optimization parameters specified in the structure options.

x = fminbnd(fun,x1,x2,options,P1,P2,...) provides for additional arguments, P1, P2, etc., which are passed to the objective function, fun. Use options=[] as a placeholder if no options are set.

[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.

Arguments

Input Arguments. Table 4-1, Input Arguments, contains general descriptions of arguments passed in to fminbnd. This section provides function-specific details for fun and options:

fun
The function to be minimized. fun is 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.
x = fminbnd(@myfun,x0)
where myfun is a MATLAB function such as
function f = myfun(x) f = ... % Compute function value at x.
fun can also be an inline object.
x = fminbnd(inline('sin(x*x)'),x0);

options
Options provides the function-specific details for the options parameters.

`fun`	The function to be minimized. `fun` is 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. x = fminbnd(@myfun,x0) where `myfun` is a MATLAB function such as function f = myfun(x) f = ... % Compute function value at x. `fun` can also be an inline object. x = fminbnd(inline('sin(x*x)'),x0);
`options`	Options provides the function-specific details for the `options` parameters.

Output Arguments. Table 4-2, Output Arguments, contains general descriptions of arguments returned by fminbnd. This section provides function-specific details for exitflag and output:

exitflag
Describes the exit condition:

> 0
The function converged to a solution x.

0
The maximum number of function evaluations or iterations was exceeded.

< 0
The function did not converge to a solution.

output
Structure containing information about the optimization. The fields of the structure are:

iterations
Number of iterations taken.

funcCount
Number of function evaluations.

algorithm
Algorithm used.

`exitflag`	Describes the exit condition:
	`> 0`	The function converged to a solution `x`.
	`0`	The maximum number of function evaluations or iterations was exceeded.
	`< 0`	The function did not converge to a solution.
`output`	Structure containing information about the optimization. The fields of the structure are:
	`iterations`	Number of iterations taken.
	`funcCount`	Number of function evaluations.
	`algorithm`	Algorithm used.

Options

Optimization options parameters used by fminbnd. You can use optimset to set or change the values of these fields in the parameters structure, options. See Table 4-3, Optimization Options Parameters,, 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) dislays output only if the function does not converge.

MaxFunEvals
Maximum number of function evaluations allowed.

MaxIter
Maximum number of iterations allowed.

TolX
Termination tolerance on x.

`Display`	Level of display. `'off'` displays no output; `'iter'` displays output at each iteration; `'final'` displays just the final output; `'notify'` (default) dislays output only if the function does not converge.
`MaxFunEvals`	Maximum number of function evaluations allowed.
`MaxIter`	Maximum number of iterations allowed.
`TolX`	Termination tolerance on `x`.

Examples

A minimum of 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

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

Algorithm

fminbnd is an M-file. The algorithm is based on Golden Section search and parabolic interpolation. A Fortran program implementing the same algorithm is given in [1].

Limitations

The function to be minimized must be continuous. fminbnd may 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.

See Also

@ (function_handle), fminsearch, fmincon, fminunc, optimset, inline

References

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

fgoalattain fmincon