extremere

Extremere is a term used in optimization theory to denote a class of extremal points associated with a primary objective under a designated family of constraints. In contrast to a simple maximum or minimum, an Extremere captures a point that achieves an extreme value in the principal criterion while satisfying secondary and possibly multiple criteria. The concept is used to analyze trade-offs in multi-criteria problems and to describe boundary behavior of feasible regions.

Formally, let F ⊆ R^n be a feasible set and f: F → R a continuous objective. An x* ∈ F is an Extremere if there exists a constraint G(x) ≤ c such that x* maximizes f on the subset {x ∈ F | G(x) ≤ c}, and the constraint is active at x* (G(x*) = c). This framework extends single-objective extrema to settings with constrained extremality and multi-objective considerations.

Existence and properties: under compactness of F and continuity of f, Extremeres exist. Uniqueness is not guaranteed, particularly in non-convex or multi-objective problems. Extremeres are commonly analyzed using Karush–Kuhn–Tucker (KKT) conditions and through the geometry of supporting hyperplanes; they often align with boundary corners or ridges where active constraints balance the objective gradient.

Examples and applications: in linear programming, maximizing a linear objective over a polygon yields Extremeres at vertices. In nonlinear problems, several Extremeres may occur along ridges or at boundary segments. The term is used in theoretical discussions of trade-offs, design optimization, and economic models to describe points that dominate under a primary criterion while respecting secondary requirements.

Etymology and usage: the coinage blends "extreme" with the suffix -ere and has appeared in specialized optimization literature to emphasize dual extremal behavior. See also extrema, Pareto optimality, and Lagrange multiplier methods.