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.

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.