Arg max
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In mathematics, arg max (or argmax) stands for the argument of the maximum, that is to say, the value of the given argument for which the value of the given expression attains its maximum value. This is well-defined only if the maximum is reached at a single value. Thus
- <math>x_0 = \arg \max_x f(x)<math>
holds if and only if x0 is the unique value of x for which f(x) is maximized. So, for example,
- <math>\arg \max_{x \in \Bbb{R}} x(10-x) = 5<math>
since the maximum value of x(10 − x) is 25, which happens when x = 5.
However, in case the maximum is reached at many values, arg max can be extended to value a set of solutions.
Then, we have for example
- <math>\arg \max_{x \in [0;4\pi]} \cos(x) = \{0;2\pi;4\pi\}<math>
since the maximum value of cos(x) is 1, which happens when x = 0, 2π or 4π.
arg min (or argmin) is defined analogously.