optimal

Optimal is an adjective used to describe the best or most favorable outcome under a given set of constraints. In optimization, an optimal solution is a value of the decision variables that yields the best possible value of an objective function within the feasible region. Depending on the problem, the optimum can be global (the best among all feasible points) or local (best within a neighborhood).

Identifying optima typically involves conditions known as optimality conditions. For constrained problems, methods such as Lagrange multipliers and the Karush-Kuhn-Tucker (KKT) conditions are widely used. When the objective and feasible set form a convex problem, any local optimum is also global, enabling efficient solution via convex optimization techniques. In non-convex problems, multiple local optima may exist and finding the global optimum can be more challenging.

Optimal solutions are central to many disciplines. In operations research and economics, they underpin resource allocation and policy design. In engineering and control theory, optimal control problems seek input signals that maximize performance over time. In computer science and machine learning, optimization is used to train models by minimizing loss or maximizing utility. Real-world applications include routing, scheduling, design optimization, and portfolio optimization.

Terminology: “optimal” is an adjective; “optimum” is a noun or adjective meaning the best possible value; “optimality” denotes the property of being optimal.

See also: optimization, convex optimization, linear programming, optimal control, KKT conditions.