Lagrange Multipliers for Constrained Quadratic Optimization
Calculus & Optimization DS practice problem on Onlearn.
Difficulty: hard.
Topics: Lagrange Multipliers for Constrained Quadratic Optimization, Lagrangian Multipliers, Hessian Matrix, Block Matrix Systems, Linear Equality Constraints, Dual Variables, Linear Algebra, Multivariate Calculus, Constrained Optimization, Numerical Analysis, Convex Geometry, Matrix Inversion, Stationary Points, KKT Conditions, Gradient Descent Foundations, Quadratic Forms.
Given a quadratic objective function f(x) = 1/2 x^T Q x + c^T x subject to the linear equality constraint A x = b, implement a function that solves for the optimal vector x using the Lagrange Multiplier method by solving the KKT system: [[Q, A^T], [A, 0]] [x, λ]^T = [ c, b]^T.