Convex Optimization: Duality and KKT Conditions

This lecture covers the duality concept in convex optimization, focusing on the second-order cone programming (SOCP) dual via conic duality and the Karush-Kuhn-Tucker (KKT) conditions for sums of logarithms. It explains the formulation of SOCP in conic form, the Lagrangian setup, and the dual objective function evaluation. Additionally, it explores the optimality conditions for equality constrained least squares and the suboptimality of a simple covering ellipsoid. The application of Farkas' lemma in financial markets to detect arbitrage opportunities is also discussed, emphasizing the importance of expected payoff matching security prices.