Difference between sup, inf and min, max

In the past, the decision goals for optimizing functions were always either minimizing or maximizing. Recently, while reading papers, I have noticed that in many high-quality papers, the notation “inf” or “sup” is often used.

“inf” is short for infimum, and “sup” is short for supremum.

Using "inf" or "sup" always ensures the existence of the infimum or supremum of a function, while the minimum or maximum of a function may sometimes not exist.

For example, consider the graph of the function \(f(x)=\sin(x)/x\):

Python codes for this picture:

import matplotlib.pyplot as plt
import numpy as np

x = np.arange(-10, 10, 0.1)
plt.plot(x, np.sin(x)/x)

# facecolor generates a hollow circle, s is the size
plt.scatter(0, 1, facecolor = 'none', edgecolor = 'r', s = 100)
plt.show()

This function is undefined at \(x=0\), so its maximum value, or “max,” does not exist. However, we can observe that the smallest upper bound of \(f(x)\) is 1 (any value not less than its maximum value is an upper bound), i.e., \(\sup f(x)=1\).

  • The definition of “sup”: The smallest upper bound of a set.
  • The definition of “inf”: The largest lower bound of a set.



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