On the Definition of a Minimum in Parameter Optimization

Abstract

For the unconstrained minimization of an ordinary function, there are essentially two definitions of a minimum. The first involves the inequality $\le$, and the second, the inequality <. The purpose of this Forum is to discuss the consequences of using these definitions for finding local and global minima of the constant objective function. The first definition says that every point on the constant function is a local minimum and maximum, as well as a global minimum and maximum. This is not a rational result. On the other hand, the second definition says that the constant function cannot be minimized in an unconstrained problem. It must be treated as a constrained problem where the constant function is the lower boundary of the feasible region. This is a rational result. As a consequence, it is recommended that the standard definition ($\le$) for a minimum be replaced by the second definition (<).