MATLAB:Fminbnd and fminsearch

The fminbnd command in MATLAB can be used to find the value of a single parameter of a multivariable 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 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. In addition to the examples below, there is another page with more examples at MATLAB:Fminbnd/Examples.

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.

Plot showing the function on the domain between -10 and 10

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

\(f(x, y, z)=\frac{x}{10}+\cos(x)+\sin(y)+z\,\!\)

We will assume that two of the three variables (\(y\) and \(z\)) are known and that we want to find \(x\) given those values. We will further assume that \(y=1\) and \(z=\pi\). A graph of this function for values of \(x\) 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 fzero 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.

Plot focusing on domain between 0 and 5 showing minimum at 3.0414

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 \(x=0\) and \(x=5\).

Plot focusing on domain between 5 and 7 showing minimum at 5.0000

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 \(x\) 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.

Plot focusing on domain between -9 and 10 showing "minimum" at 3.0414

Plot focusing on domain between -4 and -3 showing minimum at -3.2418

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 \(x=-3.2418\), 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 \(x=-9\), where \(f=2.1719\); since MATLAB was able to find at least one local minimum, however, it did not check the boundary values.