Key terms in optimization include the objective function, which quantifies the performance or cost of a solution, and constraints, which define the permissible range of solutions. For example, in linear programming, the objective function is linear, and constraints are also linear inequalities or equations. Nonlinear programming extends this to nonlinear functions, introducing complexities like local optima and multiple solutions.
Another fundamental concept is the feasible region, representing all possible solutions that satisfy the constraints. The optimal solution lies within this region, either at a vertex (in linear problems) or elsewhere (in nonlinear cases). Duality is a critical principle in optimization, where a primary problem is paired with a dual problem, often providing insights or alternative solution methods.
Common optimization algorithms include gradient descent, which iteratively refines solutions by moving in the direction of steepest descent, and simulated annealing, inspired by metallurgy, which explores solutions probabilistically to avoid local optima. Metaheuristics, such as genetic algorithms and particle swarm optimization, are used for complex, high-dimensional problems where traditional methods may fail.
Optimizationstermerna also encompasses terms like convexity, where the objective function’s feasible region is convex, guaranteeing a single global optimum, and sensitivity analysis, which examines how changes in constraints or parameters affect the solution. Stochastic optimization addresses problems with uncertain or probabilistic data, while robust optimization ensures solutions remain effective despite variability.
Modern applications extend to machine learning, where optimization drives model training, and reinforcement learning, where policies are optimized through interaction with an environment. Understanding these terms is essential for formulating, solving, and interpreting optimization problems across disciplines.