optimointiongelman

Optimointiongelma, or optimization problem, is a formal task in mathematics and applied sciences that seeks to identify the best feasible solution from a set of candidates, according to an objective function. It typically involves decision variables, an objective to be minimized or maximized, and a set of constraints that define the feasible region.

A standard formulation includes an objective function f(x) to optimize, a decision vector x, and a collection of constraints such as equalities or inequalities that restrict x. The goal is to find a point x within the feasible region that yields the best value of f. Problems can be continuous or discrete, single-objective or multi-objective, and linear or nonlinear. Special classes such as linear programming focus on linear f and linear constraints, while convex optimization deals with convex objective functions and convex feasible sets, offering guarantees about global optima under certain conditions.

Solution methods vary by problem type. Exact methods include linear programming, integer programming, quadratic programming, and dynamic programming, which aim to find the global optimum. Approximate or heuristic methods—such as local search, genetic algorithms, simulated annealing, and various metaheuristics—are used for complex or large-scale problems where exact solutions are impractical. Many general problems are computationally hard (often NP-hard), but certain subclasses admit polynomial-time algorithms.

Optimization has wide applications across operations research, manufacturing, logistics, finance, energy systems, and machine learning, where performance criteria, costs, or risks are minimized or maximized. The field has a long history, with key developments in linear programming and convex optimization contributing to practical decision-making and theoretical foundations.