Floordmin

Floordmin is a mathematical operator described as a simple combination of the floor function and the minimum operator. For two real numbers x and y, floordmin(x, y) is defined as floor(min(x, y)). Because the floor function is monotone, floordmin(x, y) is also equal to min(floor(x), floor(y)).

Generalization to more operands is straightforward. For real numbers a1, a2, ..., an, floordmin(a1, a2, ..., an) is floor(min(a1, a2, ..., an)), which is equivalent to min(floor(a1), floor(a2), ..., floor(an)).

Key properties include its dependence on both floor and min, yielding an integer result. It is commutative with respect to its real-valued arguments, since min is commutative and flooring either way preserves that symmetry. The operation is useful when a discretized, integer-valued outcome is required from continuous inputs, because it effectively rounds down after considering the smallest input value.

Examples illustrate the idea. If x = 5.7 and y = 3.2, floordmin(x, y) = floor(min(5.7, 3.2)) = floor(3.2) = 3. This also matches min(floor(5.7), floor(3.2)) = min(5, 3) = 3.

Applications of floordmin appear in discretization methods within optimization, scheduling, and resource allocation, where a nonnegative integer outcome is needed and decisions are based on the smallest continuous bound. See also floor function and the min operator for related concepts.