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

The arguments passed into the function are described in Table 1-1. The arguments returned by the function are described in Table 1-2. Details relevant to fminbnd are included below for fun, options, exitflag, and output.


`fun`	The function to be minimized. `fun` takes a scalar `x` and returns a scalar value `f` of the objective function evaluated at `x`. You can specify `fun` to be an inline object. For example, x = fminbnd(inline('sin(x*x)'),x0) Alternatively, `fun` can be a string containing the name of a function (an M-file, a built-in function, or a MEX-file). If `fun='myfun'` then the M-file function `myfun.m` would have the form function f = myfun(x) f = ... % Compute function value at x
`options`	Optimization parameter options. You can set or change the values of these parameters using the `optimset` function. `fminbnd` uses these `options` structure fields: `Display` - Level of display. `'off'` displays no output; `'iter'` displays output at each iteration; `'final'` displays just the final output. `MaxFunEvals` - Maximum number of function evaluations allowed. `MaxIter` - Maximum number of iterations allowed. `TolX` - Termination tolerance on `x`.
`exitflag`	Describes the exit condition: `> 0` indicates that the function converged to a solution `x`. `0` indicates that the maximum number of function evaluations or iterations was reached. `< 0` indicates that the function did not converge to a solution.
`output`	A structure whose fields contain information about the optimization: `output.iterations` - The number of iterations taken. `output.algorithm` - The algorithm used. `output.funcCount` - The number of function evaluations.

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

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.