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An optimization problem seeks an input that minimizes or maximizes a real-valued objective function over a feasible set defined by the problem's constraints.
Mathematical optimization is the study of selecting the best feasible solution from a set of alternatives by maximizing or minimizing an objective function. An optimization problem typically consists of a search space, candidate or feasible solutions, an objective function, and constraints that restrict which solutions are allowed. Problems may involve continuous or discrete variables, and they can be unconstrained, constrained, convex, nonconvex, deterministic, or stochastic. Because maximizing a function is equivalent to minimizing its negative, many problems are formulated as minimization tasks. A central distinction is between local and global optimality. A local optimum is at least as good as nearby feasible solutions, whereas a global optimum is at least as good as every feasible solution. Convexity makes optimization easier because, under appropriate conditions, every interior local minimum is also global. Optimality can be analyzed using first-order conditions, such as zero gradients and Lagrange multipliers, and second-order conditions involving the Hessian matrix. Major areas include linear, integer, nonlinear, quadratic, conic, stochastic, robust, combinatorial, dynamic, and multi-objective optimization. Applications range from engineering, economics, machine learning, and operations research to control, biology, finance, and scientific modeling.
An optimization problem seeks an input that minimizes or maximizes a real-valued objective function over a feasible set defined by the problem's constraints.
Optimization variables may be continuous or discrete, and problems may include equality constraints, inequality constraints, uncertainty, multiple objectives, or multiple local optima.
A local optimum is best within a neighborhood, while a global optimum is best over the entire feasible set; nonconvex problems may contain many local optima.
Convex optimization is especially important because suitable convexity conditions can ensure that a local optimum is also a global optimum.
First-order conditions identify candidate optima, while second-order conditions help classify them as minima, maxima, or saddle points.
Major techniques and subfields include linear programming, integer programming, nonlinear programming, stochastic and robust optimization, dynamic programming, optimal control, and heuristic search.
A function whose value is minimized or maximized in an optimization problem.
A candidate solution that satisfies all constraints of an optimization problem.
The set of all possible candidate solutions from which feasible solutions are selected.
A feasible point whose objective value is at least as favorable as those of sufficiently nearby feasible points.
A feasible point whose objective value is at least as favorable as that of every feasible point.
Optimization involving a convex feasible set and a convex objective for minimization or a concave objective for maximization.
A necessary condition for an interior optimum that commonly requires the objective gradient to be zero.
A matrix of second derivatives used to classify stationary points and assess local curvature.
A solution in multi-objective optimization that cannot be improved in one objective without worsening at least one other objective.
A problem that seeks any solution satisfying the constraints without optimizing an objective value.
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