The lecture proves Karush-Kuhn-Tucker (KKT) conditions, first-order optimality conditions for constrained optimization (g=0, h≥0). A seemingly complex dot product inequality simplifies to a straightforward condition on dual variable μ (μ≥0 and μh=0). These KKT conditions, along with primal feasibility, provide necessary and sufficient conditions for critical points, extending Lagrange multipliers to inequality constraints. The next lecture will cover algorithms to find these points. This segment highlights the crucial role of a symmetric positive definite matrix in the derivation of a key inequality. The speaker explains how the properties of this matrix (symmetry and positive definiteness) allow for the manipulation of an inequality without changing its direction, leading to the conclusion that the dual variable (mu star) is greater than or equal to zero. This is a significant step in simplifying a complex optimality condition.