Solmaximum

Solmaximum is a term used in a theoretical framework of optimization to designate the maximal value attained by a specified objective function over a feasible set defined by constraints. It emphasizes the focus on a particular objective or variable of interest within a constrained system, and is analogous to the standard notion of a maximum in optimization.

Formal definition: Let F ⊆ R^n be the feasible set determined by constraint relations A x ≤ b, equality constraints, or other side conditions. Let f: F → R be a real-valued objective function. The solmaximum of f on F is the value M = max_{x∈F} f(x), provided the maximum exists. If multiple maximizers exist, the set of solmaximizers is { x ∈ F : f(x) = M }. If the maximum is not attained, some authors refer to the solsupremum instead.

Computation and properties: Existence of a solmaximum depends on compactness of the feasible set and continuity of the objective. In linear programming, a nonempty, bounded feasible region guarantees existence and allows computation via standard LP methods. In nonconvex or unbounded cases, a solmaximum may not exist or may be infinite. For problems with multiple objectives, a single solmaximum may not capture all trade-offs, and one may study Pareto optimality or use scalarization techniques.

Etymology and usage: The coinage solmaximum combines “solution” and “maximum” and is used to highlight the maximization process over the feasible solution set. It is not universally standardized and is often interchangeable with conventional terms such as maximum, maximum value, or argmax in many contexts.

See also: maximum, argmax, supremum, constrained optimization, feasible set, Pareto frontier.