solmaximizers

Solmaximizers are the elements of a feasible set at which an objective function attains its maximum. The term is used in optimization literature to refer to the set of maximizing solutions to a problem and is closely related to the concept of the argmax. In many contexts, solmaximizers are described simply as the optimal solutions of a given optimization problem.

Formally, let X be a feasible set, a subset of real vector space, and f: X -> R be an objective function. The solmaximizers are the set {x in X : f(x) = max_{y in X} f(y)}. If the maximum is not attained, this set is empty. When the maximum is attained, this set is often referred to as the argmax of f over X.

Existence of solmaximizers depends on the problem data. If X is compact and f is continuous, a maximum exists by the Weierstrass theorem. Without compactness or continuity, a maximum may fail to exist; in such cases only the supremum may be identified, not an actual maximizer.

Properties of solmaximizers follow from the structure of the problem. If X is convex and f is concave, the solmaximizers form a convex set. If f is strictly concave on a convex X, the solmaximizers are unique. For differentiable f in the interior of X, first-order conditions (such as gradient vanishing in unconstrained problems or the Karush-Kuhn-Tucker conditions for constrained ones) characterize candidates for solmaximizers.

Computationally, solmaximizers are the target of various algorithms, including gradient ascent for differentiable problems and specialized convex or discrete optimization methods for structured cases.