optimalan

Optimalan is a term encountered primarily in theoretical optimization to describe a limiting or normalized optimal value associated with a sequence of optimization problems as their size grows. It is not a standard term in mainstream mathematical nomenclature, but it appears in niche papers and discussions that examine the asymptotic behavior of optimizers.

Definition and concept

Consider a sequence of problems P_n with objective functions f_n and feasible sets F_n, and a positive normalizing sequence a_n. If, for each n, a minimizer x_n^* ∈ F_n exists and the limit lim_{n→∞} f_n(x_n^*) / a_n exists, this limit is described as the optimalan of the sequence (n). In some uses, optimalan refers to the limiting object itself (the limit value) rather than solely the scalar limit.

Properties

The existence of an optimalan depends on the chosen normalization a_n and the structural properties of the problems, such as convexity, continuity, and concentration phenomena. Different normalizations can yield different optimalan values, so the concept is inherently tied to how problem size is scaled.

Applications

Optimalan is used to study scalability and asymptotic behavior in areas like combinatorial optimization, network design, and scheduling. For instance, in certain large random graphs, the optimum objective may grow proportionally with problem size, and the proportional limit can be described in terms of an optimalan.

Relationship to related concepts

Optimalan relates to asymptotic analysis, limit values, and scaling laws. It is distinct from standard limiting optimum values because it explicitly depends on a normalization sequence and is not universally standardized across disciplines.

See also

Asymptotic analysis, limit value, scaling laws, optimization theory.