# MATLAB:Fminbnd and fminsearch

## Introduction

This page discusses two different ways of getting MATLAB to find the minimum of a function (versus a data set) - **fminbnd** and **fminsearch**. The fminbnd command can find a *single* independent value that will minimize a one-dimensional function over a specific domain. The fminsearch command can find a single **vector** of values that will minimize a multi-dimensional function given some initial guess.

## fminbnd

The `fminbnd`

command in MATLAB can be used to find the value of a single parameter of a function that will minimize the value of the function on some bounded domain. The command can *only* find one minimum at a time and can only find minima based on one variable at a time. If there is a single local minimum over the domain, `fminbnd`

should find it. If there are several, it should find one of them, though it may not find the most minimum of the minima. If there is no local minimum in the range, `fminbnd`

will return one of the boundary values, depending on where the function is at its minimum value for the domain.
There are several different ways to present `fminbnd`

with the specific function and variable.

### Examples

#### Different Function Calls

##### Most General Case

The following examples show a method that will work regardless of how many input variables your function has or how you define your function - whether it is built-in, a variable containing an anonymous function, an anonymous function generated on the fly, or a .m file function. The paradigm is:

```
[INDEP_AT_MIN, FUNCTION_MIN] = fminbnd(@(DUMMY_VAR) FUNCTION_THING, LEFT_INDEP, RIGHT_INDEP)
```

where

- INDEP_AT_MIN is the calculated value of the requested variable when the function is at a minimum in the domain
- FUNCTION_MIN is the minimum value of the function at the INDEP_AT_MIN location
- DUMMY_VAR is the variable you want to use in this FUNCTION_THING to indicate which of the various inputs
`fminbnd`

is allowed to alter - FUNCTION_THING can be a built-in function, the name of an anonymous function, the name of a .m file function, or a calculation - whatever it is that you are trying to minimize; note that DUMMY_VAR must appear somewhere in this expression for
`fminbnd`

to be able to do anything - LEFT_INDEP, RIGHT_INDEP represent the minimum and maximum value of the range over which you want to minimize the function.

The example is based on wanting to determine minimum (and later, maximum) values of:

We will assume that two of the three variables ( and ) are known and that we want to find given those values. We will further assume that and . A graph of this function for values of between -10 and 10 is shown at right.

###### Function on the Fly

If you only want to solve this problem once, and the calculation only requires one line of code, the process with the least "overhead" involves creating the function on the fly inside the `fminbnd`

command. All you have to do is put your expression in for the FUNCTION_THING, making sure the DUMMY_VAR is appropriately used. You will also need to specify the boundary values.

The line:

```
[xValue, fValue] = fminbnd(@(xDummy) xDummy/10 + cos(xDummy) + sin(1) + pi, 0, 5)
```

will produce

```
xValue =
3.0414e+00
fValue =
3.2922e+00
```

which has found the local minimum between and .

Running the code:

```
[xValue, fValue] = fminbnd(@(xDummy) xDummy/10 + cos(xDummy) + sin(1) + pi, 5, 7)
```

will produce

```
xValue =
5.0000e+00
fValue =
4.7668e+00
```

since, for values between 5 and 7, there is no local minimum. The value of the function at 5 is less than the value of the function at 7, so the 5 is returned.

Finally, for

```
[xValue, fValue] = fminbnd(@(xDummy) xDummy/10 + cos(xDummy) + sin(1) + pi, -9, 10)
```

MATLAB returns:

```
xValue =
3.0414e+00
fValue =
3.2922e+00
```

Note that this is not actually the "minimum" minimum - that would be at , where the function is 2.6639, as given by

```
[xValue, fValue] = fminbnd(@(xDummy) xDummy/10 + cos(xDummy) + sin(1) + pi, -4, -2)
```

Furthermore, the actual minimum value of the function over the domain from -9 to 10 is given at , where ; since MATLAB was able to find at least one local minimum, however, it did not check the boundary values.

A slightly "longer" version of this would have you pre-define the and variables first, then use the variables in the function definition: expression in for the FUNCTION_THING, making sure the DUMMY_VAR is appropriately used. The line:

```
y = 1;
z = pi;
[xValue, fValue] = fminbnd(@(xDummy) xDummy/10 + cos(xDummy) + sin(y) + z, -4, -2)
```

This works the same as the last example above - recall that when anonymous functions are created, they can take "snapshots" of the workspace. In the case above, the FUNCTION_THING is able to "see" what the variables `y`

and `z`

contain.

###### Pre-Defined Functions

If you want to solve the same problem multiple times, perhaps by altering initial guesses or altering one or more of the constant parameters in the function, you should first define the function. If the expression can be calculated all on one line, you may choose to use a variable with an anonymous function - for example:

```
MyFun = @(xa, ya, za) xa/10 + cos(xa) + sin(ya) + za
```

On the other hand, if the expression is more complicated or if you just want to put it in a .m file, you can certainly do that as well. For example, instead of the `MyFun`

variable above, you could write a `MyFun.m`

file containing:

```
function out = MyFun(xb, yb, zb)
out = xb/10 + cos(xb) + sin(yb) + zb;
```

In either case, you would use `fminbnd`

by properly building the FUNCTION_THING using the name of the function. As in the previous example, you can hard-code the values for the and values:

```
[xValue, fValue] = fminbnd(@(xDummy) MyFun(xDummy, 1, pi), -4, 2)
```

or you can use variables to transmit the information:

```
y = 1;
z = pi;
[xValue, fValue] = fminbnd(@(xDummy) MyFun(xDummy, y, z), -4, 2)
```

In any event, note again that the FUNCTION_THING only has one unknown, and that unknown is named as the DUMMY_VAR.

##### Single-Variable Anonymous Functions

For single-variable anonymous functions, there is a slightly simpler way to run `fminbnd`

- all you need to do is give the name of the variable to FUNCTION_THING rather than setting up the DUMMY_VAR. For example, to find the minimum value of on the domain from going from -2 to 3, you can set up an anonymous function for it as:

```
MyAnonCalc = @(x) x.^2-cos(x)
```

and then solve with:

```
[xValue, fValue] = fminbnd(MyAnonCalc, -2, 3)
```

to get

```
xValue =
-4.5124e-07
fValue =
-1.0000e+00
```

##### Single-Variable Built-in Functions and .m Functions

The following method works for both built-in single-input functions and .m file single-input functions (i.e. **not** variables containing anonymous functions). The first argument is the name of the function in single quotes and the second argument is the initial guess or initial bracket for the one variable of the function. For example, to find the minimum value of for the domain of going from 0 to 4, you can type

```
[xValue, fValue] = fminbnd('cos', 0, 4)
```

and get

```
xValue =
3.1416e+00
fValue =
-1.0000e+00
```

If you have a .m file function of one variable, the process is the same. For example, to find the minimum value of on a domain from -2 to 3, you could first create a .m file called `MyCalc.m`

with:

```
function out = MyCalc(in)
out = in.^2 - cos(in);
```

then use fminbnd:

```
[xValue, fValue] = fminbnd('MyCalc', -2, 3)
```

and get

```
xValue =
-4.5124e-07
fValue =
-1.0000e+00
```

#### Finding Maximum Values

There is no `fmaxbnd`

command. Conveniently, all you need to do to figure out the value and location of the maximum values would be to give `fminbnd`

your function multiplied by -1. The only problem here is that the value of the function `fminbnd`

returns will be the opposite of what you want. For example:

```
[xValue, fValueSignWrong] = fminbnd(@(xDummy) -(xDummy/10 + cos(xDummy) + sin(1) + pi), 5, 7)
```

will return

```
xValue =
6.3834e+00
fValueSignWrong =
-5.6164e+00
```

in order to get the actual value of the function at a maximum, you would need to re-run the calculation or just create a new variable that is equal to -1 multiplied by the function value MATLAB gave:

```
[xValue, fValueSignWrong] = fminbnd(@(xDummy) -(xDummy/10 + cos(xDummy) + sin(1) + pi), 5, 7)
fValue = -1*fValueSignWrong
```

to get

```
xValue =
6.3834e+00
fValueSignWrong =
-5.6164e+00
fValue =
5.6164e+00
```

### Where the Mins Are

As noted above, if you give MATLAB a domain with multiple local minima, it will not always find the most-minimum minima; here are two graphs showing 19,900 runs of `fminbnd`

with different boundaries. The x axis represents the left boundary and the y axis represents the right boundary.

The black regoins indicate left and right values that will not work since "Right" would be to the left of "Left" there. There are four local minima over the entire domain from -10 to 10. In the left graph, with the locations, these are represented by

- Dark blue: (-9.5250, 2.0356)
- Cyan: (-3.2418, 2.6639)
- Yellow: (3.0414, 3.2922)
- Red: (9.4246, 3.9205)

In the right graph, with the values of the minimum, these are represented by:

- Dark blue: (-9.5250, 2.0356)
- Medium blue: (-3.2418, 2.6639)
- Cyan: (3.0414, 3.2922)
- Green: (9.4246, 3.9205)

For large enough - or well-placed enough - domains, MATALAB will find one of the local minima; if the domain does not span a minimum, it will report back the minimum value at either the left or right boundary. If the domain spans multiple minima, it will report back one of the minima, though not always the minimum one. Notice when the domain is [-10 -4] the dark blue minimum is found, but when the domain is [-10 0], the cyan/medium blue minimum at -3.2418 is found -- even though its value (2.6639) is greater than the value at -9.5250. These graphs show that you need to make sure your domain is large enough to include the minimum you are trying to find but not so large as to include multiple minima.

The code for making these graphs is given in the expandable box here; click the "Expand" link at right.

```
clear; format short e
MyFun = @(xa, ya, za) xa/10 + cos(xa) + sin(ya) + za
N = 200;
[xl, xr] = meshgrid(linspace(-10, 10, N));
MyMinLoc = zeros(size(xl))-20;
MyMinVal = MyMinLoc;
for k=1:(N-1)
for l=(k+1):N
[MyMinLoc(l, k), MyMinVal(l, k)] = ...
fminbnd(@(xD) MyFun(xD, 1, pi), xl(l, k), xr(l, k));
end
end
MyMap = jet;
figure(1); clf
%surfc(xl, xr, MyMinLoc)
imagesc(xl(1,:), xr(:,1), MyMinLoc, [-10 10])
set(gca, 'YDir', 'Normal'), xlabel('Left'), ylabel('Right')
title('Locations of Minimum Found by fminbnd')
view(2), shading interp
colormap([0 0 0; MyMap]); BarH = colorbar;
set(BarH, 'YLim', [-10 max(MyMinLoc(:))])
print -dpng -r0 BasinLocations
figure(2); clf
%surfc(xl, xr, MyMinLoc)
imagesc(xl(1,:), xr(:,1), MyMinVal, [2, 5.7])
set(gca, 'YDir', 'Normal'), xlabel('Left'), ylabel('Right')
title('Values of Minimum Found by fminbnd')
view(2), shading interp
colormap([0 0 0; MyMap]); BarH = colorbar;
set(BarH, 'YLim', [2 5.6])
print -dpng -r0 BasinValues
```

## fminsearch

The `fminsearch`

command in MATLAB can be used to find the value of a single *vector* input of a multivariable function that will minimize the value of the function on some *unbounded* domain. The command can only find one minimum at a time and can only find minima based on one variable at a time - but that variable can be a vector instead of a single entry.

### Most General Case

The following examples show a method that will work regardless of how many input variables your function has or how you define your function - whether it is built-in, a variable containing an anonymous function, an anonymous function generated on the fly, or a .m file function. The paradigm is:

```
[INDEP_AT_MIN, FUNCTION_MIN] = fminsearch(@(DUMMY_VAR) FUNCTION_THING, INIT_GUESS)
```

where

- INDEP_AT_MIN is the calculated value of the requested variable when the function is at a minimum in the domain
- FUNCTION_MIN is the minimum value of the function at the INDEP_AT_MIN location
- DUMMY_VAR is the variable you want to use in this FUNCTION_THING to indicate which of the various inputs
`fminbnd`

is allowed to alter - FUNCTION_THING can be a built-in function, the name of an anonymous function, the name of a .m file function, or a calculation - whatever it is that you are trying to minimize; note that DUMMY_VAR must appear somewhere in this expression for
`fminbnd`

to be able to do anything - INIT_GUESS represents an initial guess for where the minimum value is on the surface. There need to be as many values in the initial guess as there are entries in the dummy variable.

### Examples

#### One Value

If you want to find the location of the minimum of some function where is a single value (a scalar), all you need is a function and an initial guess. For example, to find the minimum of:

you can run:

```
f1 = @(x) (x+3).*(x).*(x-1).*(x-3);
[xval, fval] = fminsearch(@(xDummy) f1(xDummy), 0)
```

and MATLAB will find a minimum of . Note, however, if you make an initial guess of 1:

```
[xval2, fval2] = fminsearch(@(xDummy) f1(xDummy), 1)
```

will return the local - but not global - minimum of . Since there is no way to force `fminsearch`

to find a minimum within a particular set of boundaries, either you should use `fminbnd`

if you only have one parameter or you should be extremely careful in picking your initial conditions. The code below demonstrates that MATLAB finds different locations and values for the minimum depending on initial condition:

```
clear;
f1 = @(x) (x+3).*(x).*(x-1).*(x-3);
xinit = linspace(-40, 40, 1000);
for k = 1:length(xinit)
[xval(k), fval(k)] = fminsearch(@(xDummy) f1(xDummy), xinit(k));
end
figure(1); clf
plot(xinit, xval)
xlabel('Initial x'), ylabel('Final x')
figure(2); clf
plot(xinit(fval>-20), f1(xinit(fval>-20)), 'b.', ...
xinit(fval<-20), f1(xinit(fval<-20)), 'r.')
legend('Basin for min near -11', 'Basin for min of near -30', 'location', 'best')
```

#### Multiple Value Vector

Imagine if you were to want to find the minimum of some function

which has a discernible minimum at (1, 2); you could use:

```
[rval, fval] = fminsearch(@(rDummy) (rDummy(1)-1).^2+(rDummy(2)-2).^2, [0 0])
```

to have MATLAB start at the origin and find the value of the *vector* `rDummy`

that has the *x* and *y* coordinate in it.
Since this function only has one minimum, MATLAB should be able to find it as long as the initial guess is somewhat reasonable.

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