Quadratic Programming optimizes a quadratic objective function subject to linear constraints. Standard form: minimize (1/2)xᵀQx + cᵀx subject to Ax ≤ b, x ≥ 0. If Q is positive semidefinite, the problem is convex and efficiently solvable. Solution methods: interior point algorithms, active set methods, and specialized quadratic solvers. For convex QP, KKT conditions are necessary and sufficient. Applications include: portfolio optimization (minimize risk subject to return constraints), support vector machines in machine learning, and least-squares problems with constraints. Mixed-integer quadratic programming (MIQP) combines quadratic objectives with integer variables, important for portfolio optimization with transaction constraints and feature selection.