Effective domain

In convex analysis, a branch of mathematics, the effective domain extends of the domain of a function defined for functions that take values in the extended real number line ${\displaystyle [-\infty ,\infty ]=\mathbb {R} \cup \{\pm \infty \}.}$

In convex analysis and variational analysis, a point at which some given extended real-valued function is minimized is typically sought, where such a point is called a global minimum point. The effective domain of this function is defined to be the set of all points in this function's domain at which its value is not equal to ${\displaystyle +\infty .}$^[1] It is defined this way because it is only these points that have even a remote chance of being a global minimum point. Indeed, it is common practice in these fields to set a function equal to ${\displaystyle +\infty }$ at a point specifically to exclude that point from even being considered as a potential solution (to the minimization problem).^[1] Points at which the function takes the value ${\displaystyle -\infty }$ (if any) belong to the effective domain because such points are considered acceptable solutions to the minimization problem,^[1] with the reasoning being that if such a point was not acceptable as a solution then the function would have already been set to ${\displaystyle +\infty }$ at that point instead.

When a minimum point (in ${\displaystyle X}$) of a function ${\displaystyle f:X\to [-\infty ,\infty ]}$ is to be found but ${\displaystyle f}$'s domain ${\displaystyle X}$ is a proper subset of some vector space ${\displaystyle V,}$ then it often technically useful to extend ${\displaystyle f}$ to all of ${\displaystyle V}$ by setting ${\displaystyle f(x):=+\infty }$ at every ${\displaystyle x\in V\setminus X.}$^[1] By definition, no point of ${\displaystyle V\setminus X}$ belongs to the effective domain of ${\displaystyle f,}$ which is consistent with the desire to find a minimum point of the original function ${\displaystyle f:X\to [-\infty ,\infty ]}$ rather than of the newly defined extension to all of ${\displaystyle V.}$

If the problem is instead a maximization problem (which would be clearly indicated) then the effective domain instead consists of all points in the function's domain at which it is not equal to ${\displaystyle -\infty .}$