Lineairprogrammimine

Lineairprogrammimine is a mathematical optimization technique used to obtain the best outcome under linear relationships. It models a linear objective function to be optimized (maximized or minimized) subject to linear equality and inequality constraints, with decision variables constrained to take real values within specified bounds.

The conventional formulation expresses either as maximize c^T x subject to Ax ≤ b, x ≥ 0, or in equivalent equality form with Ax = b and x ≥ 0. The feasible region is a convex polyhedron, and if an optimum exists for a bounded problem, it is often attained at a vertex (basic feasible solution). Duality associates a second problem that provides bounds and insights.

Solutions are typically obtained by the simplex method, which traverses adjacent vertices, or by interior-point methods, which traverse the interior of the feasible region and have polynomial-time worst-case performance. In practice, linear programs can be solved efficiently for large-scale instances. Sensitivity or post-optimality analysis studies how changes in c, A, b affect the solution.

Lineairprogrammimine has broad applications in operations research, economics and engineering, including resource allocation, production planning, transportation and network flows, scheduling, diet problems, and financial portfolio optimization. It serves as a relaxation technique for integer linear programming, where some variables are required to be integral.

Historical note: the method was formalized by George Dantzig in 1947, contributing to the development of operations research. While linear programming focuses on linear relationships, nonlinear problems require other techniques.