Linear programming

Linear programming optimizes a linear objective function subject to linear equality and inequality constraints. Its feasible solutions form a convex polyhedron, and an optimum occurs at an extreme point when one exists.

Feasible region: The set of points satisfying every constraint in an optimization problem. Linear constraints carve out the region in which every permissible solution lies.
Linear function: A function expressible as a constant plus a linear combination of its variables. The objective and constraint expressions must be linear in the decision variables.
Diet problem: An optimization problem that selects foods to meet nutritional requirements at minimum cost. It is a classic example of balancing linear nutrient constraints against a linear cost.
Integer programming: Optimization with some or all decision variables restricted to integer values. Discrete decisions violate the continuous-variable assumption of ordinary linear programming.
Simplex algorithm: An algorithm that solves linear programs by moving between vertices of the feasible polyhedron. It searches the vertices where a linear objective can attain its optimum.
Convex set: A set containing the entire line segment between any two of its points. Intersections of linear half-spaces form convex feasible regions.
Transportation problem: An optimization problem that minimizes shipment cost while meeting supply and demand requirements. Its structured linear program assigns quantities from sources to destinations.
Nonlinear programming: Optimization in which the objective or at least one constraint is nonlinear. Curved objectives or feasible boundaries fall outside the linear model.
Interior-point method: An optimization algorithm that approaches a solution through the interior of a feasible region. It offers a polynomial-time alternative to vertex-to-vertex simplex search.
Half-space (mathematics): One of the two regions into which a hyperplane divides a space. Each linear inequality restricts solutions to a half-space.

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