Mathematical optimization

Mathematical optimization is the study of choosing a feasible decision that maximizes or minimizes an objective function subject to constraints.

Objective function: A function that assigns a numerical value to each candidate decision. It quantifies what optimization seeks to maximize or minimize.
Gradient descent: An iterative method that updates variables opposite the gradient of an objective function. It is a standard approach for minimizing differentiable objectives.
Linear programming: Optimization of a linear objective subject to linear equality and inequality constraints. It models resource allocation when effects and limits are linear.
Convexity: A geometric property in which line segments between points in a set remain within it, or function values obey a convexity inequality. Convexity separates problems with dependable global-optimum guarantees from harder cases.
Feasible region: The set of decisions that satisfy all constraints in a mathematical problem. Optimization searches this set rather than every possible decision.
Simplex algorithm: An algorithm for linear programming that moves between vertices of a feasible polyhedron. It exploits the geometry of linear objectives and constraints.
Integer programming: Optimization problems in which some or all decision variables must be integers. It represents indivisible choices such as selecting projects or routes.
Local optimum: A feasible point whose objective value is no worse than nearby feasible points. A local optimum need not be the globally best feasible solution.
Constraint (mathematics): A condition that restricts which values variables may take in a mathematical problem. Constraints determine which candidate solutions are allowed.
Karush–Kuhn–Tucker conditions: First-order conditions that characterize candidate optima for constrained differentiable problems under regularity assumptions. They extend derivative-based optimality tests to constrained problems.

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