Kelley's Cutting Plane Method solves convex programming problems by iteratively refining a polyhedral approximation of the feasible region. The algorithm: (1) starts with an initial linear approximation; (2) solves the LP relaxation; (3) if the solution is feasible for the original problem, it's optimal; (4) otherwise, adds a cutting plane (linear constraint) that separates the current solution from the feasible region; (5) repeats until convergence. The method is particularly effective for problems with complicated convex constraints that can be separated. Modern variants include bundle methods and level methods used in large-scale optimization.