Convex Programming involves optimizing a convex objective function over a convex feasible set. A function f is convex if f(λx + (1-λ)y) ≤ λf(x) + (1-λ)f for all x, y and λ ∈ [0,1]. Key properties: any local minimum is a global minimum; first-order optimality conditions (KKT conditions) are sufficient; numerous efficient algorithms exist. Applications include: portfolio optimization, machine learning (SVM, logistic regression), signal processing, control theory, and network flow problems. Convex programming bridges linear and nonlinear optimization with computational tractability.