Optimal boundedness criteria for extended-real-valued functions

Abstract

The notion of genuinely bounded below function is introduced and characterized by means of the concept of co-equilibrated function. As an application, we state two boundedness criteria for extended-real-valued functions, both optimal in a clearly defined sense. The first one says that an extended-real-valued function minorized by an affine map and coinciding from some value up with a co-equilibrated function is bounded below. The second criterion states that an extended-real-valued function minorized by an affine map is bounded below provided that one of its sub-level sets is co-equilibrated.

Keywords:

• Bounded below function
• Growth co-direction
• Strong growth co-direction
• Co-equilibrated function
• Co-equilibrated set
• Fenchel-Legendre conjugate
• Barrier cone
• Sub-level set

Keywords:

• 2000 Mathematics Subject Classifications: 26B99
• 46N10
• 49J99