argminp

Argminp denotes the set of points p that minimize a given function f(p) over a specified domain P. The expression argmin_p f(p) returns the minimizers, meaning all p in P for which f(p) is no larger than f(q) for every q in P. If the function has a unique minimizer, the argmin is a single value; otherwise it can be a set of multiple minimizers.

Notation and scope: The subscript p indicates the variable being optimized, while the domain P defines the allowable values of p. In some texts, argmin is written without an explicit subscript, with the understanding that the domain is defined by context. The concept is closely related to the argmax operator, which returns the points that maximize a function; note that argmin_p f(p) = argmax_p -f(p).

Existence and uniqueness: If f is convex on a convex domain and strictly convex, the minimizer is unique. In general, there may be zero, one, or several minimizers. Existence is guaranteed under conditions such as a closed, bounded domain with f continuous, or coercivity of f on an unbounded domain.

Computation and examples: For differentiable f, stationary points satisfy the condition gradient f(p) = 0. In convex problems, such points are global minimizers. Examples include f(p) = (p−3)², whose argmin is p = 3, and f(p) = |p|, whose argmin is {0}.

Applications: The argmin operator is fundamental in optimization, statistics, and machine learning. Parameter estimates often take the form p* = argmin_p L(p) for a loss function L. In Bayesian contexts, the maximum a posteriori estimate corresponds to the argmin of the negative log posterior.