Interior Point Methods solve linear and nonlinear programs by traversing the interior of the feasible region rather than following boundary edges. For LP, the barrier (primal-dual) method adds logarithmic barrier function μΣln(xᵢ) to prevent boundary violations, solving a sequence of barrier problems as μ → 0. The method maintains strict feasibility and complementarity, iterating using Newton steps on KKT conditions. Advantages over simplex: polynomial worst-case complexity (Karmarkar's algorithm), better performance on large sparse problems, warm-start capability. Extensions include: nonlinear programming (IPOPT), semidefinite programming, and conic optimization. Interior point methods revolutionized large-scale optimization and are implemented in all major solvers (CPLEX, Gurobi, Mosek).