optimaalien

Optimaalien is a Finnish term used in optimization theory to refer to the set of all optimal solutions or optimal values of a problem. The term derives from optimaalinen (optimal) and the plural suffix -ien, and appears in phrases such as optimaalien ratkaisujen joukko. In mathematical notation, the problem is typically stated as finding x in a feasible set X that minimizes f(x) or maximizes f(x). The collection of all x that attain the optimum value f* is the optimalien set.

The optimalien set may take different forms depending on the problem. It can be a single point, a line segment, or a higher-dimensional subset of X. Thus the optimalien set captures all decision variables that achieve the best possible objective value, not just one representative solution.

Global versus local: The global optimum is the best value over the entire feasible region, while local optima are best within a neighborhood. The global optimalien set consists of all global optima; local optima have their own local optimalien sets, which may be disjoint or form several disconnected pieces.

Notation and properties: Common notations include Argmin and Argmax, which denote the sets of minimizers and maximizers, respectively. When multiple optima exist, these operators return a set rather than a single point, reflecting the potentially multiplicative nature of the solution.

Examples: Consider maximize f(x, y) = 3x + 2y subject to x ≥ 0, y ≥ 0, x + y ≤ 4. The maximum value is 12, achieved at a single point (4, 0), so the optimalien set is that point. In contrast, maximize f(x, y) = x + y with x ≥ 0, y ≥ 0, x + y ≤ 1 yields the optimal value 1 for all points with x + y = 1 in the first quadrant, and the optimalien set is the line segment {(x, y) | x ≥ 0, y ≥ 0, x + y = 1}.

In multi-objective optimization, Pareto-optimal solutions relate to optima in a broader sense, representing trade-offs where no objective can be improved without worsening another.