These theories are traditionally divided into several main classes. Linear programming models linear relationships with linear constraints; quadratic programming adds quadratic objective terms; convex optimization deals with convex functions and sets, guaranteeing global optimality; and integer programming restricts variables to discrete values. Nonlinear and stochastic optimization extend the framework to nonconvex, dynamic, or probabilistic settings, while combinatorial optimization focuses on discrete structures such as graphs and networks.
Historical development of optimization theories began in the early twentieth century with the invention of Lagrange multipliers (analytic technique for constrained extrema) and relaxations of linear programs. The 1940s and 1950s produced the Kuhn‑Tucker conditions for nonlinear programming and the interior‑point methods of Karmarkar (1979), which revolutionized large‑scale linear problem solving. Subsequent decades added robust, global, and multi‑objective optimization, forming a rich set of tools for modern engineering demands.
Applications of optimointiteorioilla span engineering design, economics, supply‑chain planning, telecommunications, and machine‑learning hyper‑parameter tuning. In production, linear programs calculate optimal resource allocation; in portfolio theory, quadratic and convex models determine risk–return trade‑offs; and in deep learning, gradient‑based optimization implements training of neural networks. Health care benefits from integer programming in scheduling and facility location, while robust optimization protects solutions against uncertainty.
Current research in optimization theory focuses on scalability, integration of data‑driven models, and hybrid algorithms that combine exact methods with heuristics. Advances in quantum computing, evolutionary computation, and stochastic gradient descent further enrich the landscape, promising faster, more reliable solutions for increasingly complex real‑world problems.