fminbnd - Find local minimum of single-variable function on fixed interval - MATLAB (original) (raw)
Find local minimum of single-variable function on fixed interval
Syntax
Description
fminbnd is a one-dimensional minimizer that finds a local minimum for a problem specified by
x, x_1, and_x_2 are finite scalars, and_f(x) is a function that returns a scalar.
If multiple local minima exist on the interval (x1,x2),fminbnd returns only one, which is not guaranteed to be the global minimum. For details, see Local vs. Global Minimum.
[x](#bvadyg6-1-x) = fminbnd([fun](#bvadyg6-1-fun),[x1](#bvadyg6-1-x1),[x2](#bvadyg6-1-x2)) returns a value x that is a local minimizer of the scalar valued function that is described in fun in the intervalx1 < x < x2.
[x](#bvadyg6-1-x) = fminbnd([fun](#bvadyg6-1-fun),[x1](#bvadyg6-1-x1),[x2](#bvadyg6-1-x2),[options](#bvadyg6-1-options)) minimizes with the optimization options specified in options. Use optimset to set these options.
[x](#bvadyg6-1-x) = fminbnd([problem](#bvadyg6-1-problem)) minimizesproblem, where problem is a structure.
[[x](#bvadyg6-1-x),[fval](#bvadyg6-1-fval)] = fminbnd(___), for any input arguments, returns the value of the objective function computed in fun at the solution x.
[[x](#bvadyg6-1-x),[fval](#bvadyg6-1-fval),[exitflag](#bvadyg6-1-exitflag)] = fminbnd(___) additionally returns a value exitflag that describes the exit condition.
[[x](#bvadyg6-1-x),[fval](#bvadyg6-1-fval),[exitflag](#bvadyg6-1-exitflag),[output](#bvadyg6-1-output)] = fminbnd(___) additionally returns a structure output that contains information about the optimization.
Examples
Find the point where the sin(x) function takes its minimum in the range 0<x<2π.
fun = @sin; x1 = 0; x2 = 2*pi; x = fminbnd(fun,x1,x2)
To display precision, this is the same as the correct value x=3π/2.
Minimize a function that is specified by a separate function file. A function accepts a point x and returns a real scalar representing the value of the objective function at x.
Write the following function as a file, and save the file as scalarobjective.m on your MATLAB® path.
function f = scalarobjective(x) f = 0; for k = -10:10 f = f + (k+1)^2cos(kx)*exp(-k^2/2); end
Find the x that minimizes scalarobjective on the interval 1 <= x <= 3.
x = fminbnd(@scalarobjective,1,3)
Minimize a function when there is an extra parameter. The function sin(x-a) has a minimum that depends on the value of the parameter a. Create an anonymous function of x that includes the value of the parameter a. Minimize this function over the interval 0<x<2π.
a = 9/7; fun = @(x)sin(x-a); x = fminbnd(fun,1,2*pi)
This answer is correct; the theoretical value is
For more information about including extra parameters, see Parameterizing Functions.
Monitor the steps fminbnd takes to minimize the sin(x) function for 0<x<2π.
fun = @sin; x1 = 0; x2 = 2*pi; options = optimset('Display','iter'); x = fminbnd(fun,x1,x2,options)
Func-count x f(x) Procedure 1 2.39996 0.67549 initial 2 3.88322 -0.67549 golden 3 4.79993 -0.996171 golden 4 5.08984 -0.929607 parabolic 5 4.70582 -0.999978 parabolic 6 4.7118 -1 parabolic 7 4.71239 -1 parabolic 8 4.71236 -1 parabolic 9 4.71242 -1 parabolic
Optimization terminated: the current x satisfies the termination criteria using OPTIONS.TolX of 1.000000e-04
Find the location of the minimum of sin(x) and the value of the minimum for 0<x<2π.
fun = @sin; [x,fval] = fminbnd(fun,1,2*pi)
Return all information about the fminbnd solution process by requesting all outputs. Also, monitor the solution process using a plot function.
fun = @sin; x1 = 0; x2 = 2*pi; options = optimset('PlotFcns',@optimplotfval); [x,fval,exitflag,output] = fminbnd(fun,x1,x2,options)
output = struct with fields: iterations: 8 funcCount: 9 algorithm: 'golden section search, parabolic interpolation' message: 'Optimization terminated:↵ the current x satisfies the termination criteria using OPTIONS.TolX of 1.000000e-04 ↵'
Input Arguments
Function to minimize, specified as a function handle or function name.fun is a function that accepts a real scalarx and returns a real scalar f (the objective function evaluated at x).
Specify fun as a function handle for a file:
x = fminbnd(@myfun,x1,x2)
where myfun is a MATLAB® function such as
function f = myfun(x) f = ... % Compute function value at x
You can also specify fun as a function handle for an anonymous function:
x = fminbnd(@(x)norm(x)^2,x1,x2);
Example: fun = @(x)-x*exp(-3*x)
Data Types: char | function_handle | string
Lower bound, specified as a finite real scalar.
Example: x1 = -3
Data Types: double
Upper bound, specified as a finite real scalar.
Example: x2 = 5
Data Types: double
Optimization options, specified as a structure such asoptimset returns. You can use optimset to set or change the values of these fields in the options structure. See Set Optimization Options for detailed information.
| Display | Level of display (see Optimization Solver Iterative Display): 'notify' (default) displays output only if the function does not converge.'off' or'none' displays no output.'iter' displays output at each iteration.'final' displays just the final output. |
|---|---|
| FunValCheck | Check whether objective function values are valid. The default 'off' allowsfminbnd to proceed when the objective function returns a value that iscomplex orNaN. The 'on' setting throws an error when the objective function returns a value that is complex orNaN. |
| MaxFunEvals | Maximum number of function evaluations allowed, a positive integer. The default is 500. See Tolerances and Stopping Criteria. |
| MaxIter | Maximum number of iterations allowed, a positive integer. The default is 500. SeeTolerances and Stopping Criteria. |
| 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 Optimization Solver Output Functions. |
| PlotFcns | Plots various measures of progress while the algorithm executes. Select from predefined plots or write your own. Pass a function name, function handle, or a cell array of function names or handles. The default is none ([]): @optimplotx plots the current point@optimplotfunccount plots the function count@optimplotfval plots the function valueFor information on writing a custom plot function, see Optimization Solver Plot Functions. |
| TolX | Termination tolerance on x, a positive scalar. The default is 1e-4. See Tolerances and Stopping Criteria. |
Example: options = optimset('Display','iter')
Data Types: struct
Problem structure, specified as a structure with the following fields.
| Field Name | Entry |
|---|---|
| objective | Objective function |
| x1 | Left endpoint |
| x2 | Right endpoint |
| solver | 'fminbnd' |
| options | Options structure such as returned by optimset |
Data Types: struct
Output Arguments
Solution, returned as a real scalar. Typically, x is a local solution to the problem when exitflag is positive. See Local vs. Global Minimum.
Objective function value at the solution, returned as a real number. Generally, fval = fun(x).
Reason fminbnd stopped, returned as an integer.
| 1 | Function converged to a solution x. |
|---|---|
| 0 | Number of iterations exceeded options.MaxIter or number of function evaluations exceeded options.MaxFunEvals. |
| -1 | Stopped by an output function or plot function. |
| -2 | The bounds are inconsistent, meaning x1 > x2. |
Information about the optimization process, returned as a structure with fields:
| iterations | Number of iterations taken |
|---|---|
| funcCount | Number of function evaluations |
| algorithm | 'golden section search, parabolic interpolation' |
| message | Exit message |
Limitations
- The function to be minimized must be continuous.
fminbndmight only give local solutions.fminbndcan exhibit slow convergence when the solution is on a boundary of the interval.
More About
In general, optimization solvers return a local minimum (or optimum). The result might be a global minimum (or optimum), but might not.
- A local minimum of a function is a point where the function value is smaller than at nearby points, but possibly greater than at a distant point.
- A global minimum is a point where the function value is smaller than at all other feasible points.
MATLAB and Optimization Toolbox™ optimization solvers typically return a local minimum. Global Optimization Toolbox solvers can search for a global minimum, but do not guarantee that their solutions are global. For an example of global search, see Find Global or Multiple Local Minima (Global Optimization Toolbox).
Algorithms
fminbnd is a function file. The algorithm is based on golden section search and parabolic interpolation. Unless the left endpoint _x_1 is very close to the right endpoint _x_2, fminbnd never evaluates fun at the endpoints, so fun need only be defined for x in the interval _x_1 < x < _x_2.
If the minimum actually occurs at _x_1 or _x_2, fminbnd returns a point x in the interior of the interval (_x_1,_x_2) that is close to the minimizer. In this case, the distance of x from the minimizer is no more than 2*(TolX + 3*abs(x)*sqrt(eps)). See [1] or [2] for details about the algorithm.
Alternative Functionality
App
The Optimize Live Editor task provides a visual interface forfminbnd.
References
[1] Forsythe, G. E., M. A. Malcolm, and C. B. Moler. Computer Methods for Mathematical Computations. Englewood Cliffs, NJ: Prentice Hall, 1976.
[2] Brent, Richard. P. Algorithms for Minimization without Derivatives. Englewood Cliffs, NJ: Prentice-Hall, 1973.
Extended Capabilities
For C/C++ code generation:
fminbnddoes not support the problem structure argument.fminbndignores theDisplayoption and does not give iterative display or an exit message. To check solution quality, examine the exit flag.- The output structure does not include the
algorithmormessagefields. fminbndignores theOutputFcnandPlotFcnsoptions.
Version History
Introduced before R2006a