Assess students' ability to apply Lagrange multipliers and Karush–Kuhn–Tucker (KKT) conditions to solve advanced constrained optimization problems, including handling equality and inequality constraints, verifying constraint qualifications (e.g., LICQ), using complementarity slackness, applying first- and second-order optimality conditions, distinguishing local versus global optima in convex and nonconvex settings, and interpreting Lagrange/KKT multipliers; tasks may require analytical derivations, active-set identification, and evaluation of boundary solutions.