Optimization

Optimization is the process of finding the best solution from a set of feasible solutions. It involves selecting decision variables to maximize or minimize an objective function, subject to constraints. In mathematics and computer science, optimization problems are formalized as finding a point in a feasible region that optimizes a value.

Key elements include the decision variables, an objective function, and a set of constraints that define the feasible region. An optimal solution is a feasible point that yields the best objective value. A solution can be global (best over all feasible points) or local (best in a nearby neighborhood).

Common methods include exact algorithms for certain classes, such as linear programming, which uses the simplex or interior-point methods, and convex optimization, which guarantees global optima under convexity. Nonlinear programming handles nonlinear objectives or constraints. Integer programming imposes integrality, leading to combinatorial optimization problems.

When exact solutions are impractical, heuristic and metaheuristic methods—greedy algorithms, local search, genetic algorithms, simulated annealing, and swarm optimization—provide good solutions for hard problems. Stochastic optimization accounts for uncertainty.

Optimization has applications across engineering, economics, finance, data science, logistics, scheduling, and resource allocation. It also underpins machine learning model training, hyperparameter tuning, and experimental design.

Challenges include non-convexity leading to local optima, combinatorial explosion in discrete problems, and computational limits. Robust optimization and stochastic programming address uncertainty. Model formulation quality strongly influences results.