Optimums

In mathematics and optimization, an optimum is a point in the decision space where an objective function attains its best value under a given constraints set. The best value is the optimum value, and the point is an optimal solution. If the objective aims to maximize, the optimum is a maximum; if it aims to minimize, it is a minimum. The term distinguishes between the value and the input (the arg optimum).

Types: A global optimum is the best value over the entire feasible set. Local optima are best within a neighborhood and may not be global. In non-convex problems, several local optima can exist, making identification of the global optimum challenging.

Examples: The sine function has infinitely many local maxima of value 1 at x = pi/2 + 2 pi k and a global maximum of 1. More complex functions can have multiple optima or mixtures of maxima and minima. Constrained problems restrict optima to the feasible region.

Methods: Unconstrained problems can be solved by setting derivatives to zero and applying second-order tests. Constrained problems use Lagrange multipliers or KKT conditions. For non-convex problems, global optimization methods (such as branch-and-bound or global solvers) or heuristics (genetic algorithms, simulated annealing) may be used. In convex problems, any local optimum is global, simplifying computation.

Applications: Optima are central to economics, operations research, engineering, and machine learning, where the aim is the best design, policy, or model under constraints. The terms "optimal value" and "arg optimum" distinguish the value from the point.