Maximalizace

Maximalizace is the process or result of maximizing an objective function, i.e., choosing inputs to produce as large a value as possible under given constraints. The term is used in Czech and Slovak mathematics, economics and computer science. It contrasts with minimizace, the search for the smallest value.

In formal terms, a maximization problem seeks to maximize f(x) subject to x in X, where X represents feasible decisions defined by equations or inequalities. A point x* is a maximum if f(x*) ≥ f(x) for all x in X; if the inequality holds only in a neighborhood, x* is a local maximum; if it holds for all feasible x, it is a global maximum. Existence depends on continuity and compactness; convexity simplifies the search because local maxima may be global.

Common problem classes include linear programming: maximize c^T x subject to Ax ≤ b, x ≥ 0; integer programming, where some variables are restricted to integers; nonlinear programming; and dynamic programming for problems with stages. Multi-objective maximization considers trade-offs among several criteria.

Solution methods range from exact algorithms (simplex and interior-point methods for linear and convex problems; branch-and-bound, branch-and-cut for integer problems) to heuristics and metaheuristics (genetic algorithms, simulated annealing) when exact solutions are impractical. In machine learning, many learning tasks are phrased as maximization problems (e.g., likelihood or reward maximization).

Applications include economics (profit maximization), operations research (throughput or utility maximization), finance, engineering, and data science. See also minimizace and optimization.