LPPS

Linear programming problems (LPPs) are optimization problems that seek to maximize or minimize a linear objective function subject to linear constraints. In standard form for a maximization problem, one writes maximize c^T x subject to Ax ≤ b and x ≥ 0, where x is the vector of decision variables, A is a constraint matrix, and b and c are vectors. The feasible region, the set of x satisfying the constraints, is a polyhedron. If nonempty and bounded, an optimal solution, when it exists, is attained at a vertex of the polyhedron; degeneracy may yield multiple optimal solutions along an edge or face.

Solution methods: The graphical method applies only when there are two decision variables. For general problems, the simplex method moves from one basic feasible solution (vertex) to an adjacent one with an improved objective value until no improvement is possible. Interior-point methods solve the problem by traversing the interior of the feasible region via barrier or primal-dual formulations. Duality: every LP has a dual problem; strong duality states that optimal values coincide under mild conditions, and complementary slackness characterizes optimality.

Applications and properties: LPs underpin many areas of operations research, economics, and engineering, including production planning, transportation, network flows, and portfolio optimization. If the constraint matrix is totally unimodular and b is integral, optimal solutions are integral even without integer constraints. Solvers: commercial and open-source software (for example CPLEX, Gurobi, Mosek, GLPK) solve large-scale LPs efficiently.