LPløsning

LPløsning is the result of solving a linear programming problem (LP). In a typical LP, the decision variables x belong to the nonnegative orthant, and the objective is to maximize or minimize a linear function c^T x, subject to a set of linear constraints Ax ≤ b. The solution consists of an optimal value z* = c^T x* and a feasible vector x* that attains it, provided such a solution exists.

Linear programming problems model a wide range of resource allocation and optimization tasks in fields such as operations research, economics, manufacturing, and logistics. The feasible region defined by the constraints is a convex polyhedron, and, if an optimal solution exists, it is often found at an extreme point or along a face of this polyhedron. Dual variables associated with the constraints provide insight into resource prices and sensitivity.

Two principal families of algorithms are used to compute LPløsninger: the simplex method and interior-point methods. The simplex method traverses vertices of the feasible region, moving from one basic feasible solution to another, while interior-point methods search within the interior of the feasible region toward the optimum. Simplex can encounter degeneracy and multiple optimal solutions; interior-point methods are typically polynomial in theory. In practice, modern solvers combine these ideas and exploit dual information to enhance performance.

Software and practice: LPløsning problems are solved with specialized solvers such as CPLEX, Gurobi, GLPK, and COIN-OR, often via modeling languages like PuLP, Pyomo, AMPL, or JuMP. Applications include production planning, transportation, network design, energy management, and scheduling, with outputs that include the optimal decision vector x* and the optimal objective value z*.