Optimumok

Optimumok are the set of optimal solutions to an optimization problem. A single member of this set is an optimum point, the decision variable value that achieves the best possible value of the objective function under the given constraints. The term optimumok is common in Hungarian-language mathematics to refer to multiple optimal solutions; in English-language texts, authors typically use optima or optimal solutions.

Definition and scope: Given a feasible space X and an objective function f: X → R, an x* in X is a global optimum for minimization if f(x*) ≤ f(x) for all x in X, and a global optimum for maximization if f(x*) ≥ f(x) for all x in X. The collection of all such x* forms the global optimum set. In many problems, there can be more than one optimum, potentially forming a continuum. Local optima are optimal within a neighborhood: a point that is better than all nearby feasible points in some region.

Types and properties: Global optima are distinguished from local optima by the scope of comparison. In convex problems, the global optimum is unique when the objective is strictly convex; otherwise there can be infinitely many optima. In linear programming, the set of optimum points may lie on a face of the feasible region, leading to multiple optima along that face.

Applications and usage: Optimumok arise across fields such as economics, engineering, machine learning, and operations research, wherever a problem requires finding the best feasible value of an objective. Methods to identify optimumok include gradient-based techniques, Lagrange multipliers, linear and integer programming, and global optimization algorithms.

Etymology: The term reflects Hungarian usage for the plural of optimum, rooted in Latin origin, with English literature commonly using optima for the plural.