Interior point methods (Karmarkar, 1984) solve linear and convex programs by staying strictly inside the feasible region. A logarithmic barrier term −μ·Σ log(sᵢ) repels the solution from constraint boundaries, while μ is gradually reduced to zero, allowing the path to approach the boundary at the optimum. The central path traces a smooth curve from the analytic center to the optimal vertex. Unlike simplex, interior point methods run in polynomial time and scale gracefully to problems with millions of variables — the foundation of modern convex optimization solvers.