[Gauss Labs Special AI & Optimization Lecture] Optimization in General and Convex Optimization

Abstract

This lecture provides a comprehensive introduction to mathematical optimization with a particular emphasis on convex optimization and its fundamental role in modern machine learning and artificial intelligence. The presentation begins by establishing the general framework of optimization problems, including the formal mathematical formulation with objective functions, inequality constraints, and equality constraints, followed by practical examples spanning circuit design, portfolio optimization, and neural network training.

The core challenge in general optimization is highlighted through the discussion of computational complexity, where most optimization problems are extremely difficult or impossible to solve optimally. However, several important exceptions exist, including least-squares problems, linear programming, quadratic programming, and semidefinite programming, all of which share the crucial property of being convex optimization problems that can be solved efficiently with polynomial-time algorithms.

Convex optimization emerges as the central theme, distinguished by its requirement that objective and constraint functions be convex, with all equality constraints remaining linear. The lecture emphasizes why convex optimization is fundamental: it provides the theoretical foundation for many machine learning algorithms, offers guaranteed global optimality, and enables the development of reliable and efficient solution algorithms including gradient descent, Newton’s method, and interior-point methods.

The presentation systematically explores key examples of convex optimization in machine learning contexts, progressing from linear regression (equivalent to least-squares) through ridge regression and LASSO to support vector machines (SVM). Each example demonstrates how seemingly complex machine learning problems can be reformulated as convex optimization problems, enabling robust and efficient solutions. The discussion of SVM with kernels particularly illustrates how feature transformations maintain the convex structure while enabling nonlinear classification.

A significant portion of the lecture is dedicated to Lagrangian duality theory, beginning with the construction of the Lagrangian function and dual problem. The fundamental result that every dual problem is convex (regardless of whether the primal is convex) is established, along with the crucial weak and strong duality relationships. Slater’s constraint qualification is presented as a key condition ensuring strong duality in convex problems, providing both theoretical guarantees and practical algorithmic foundations.

The Karush-Kuhn-Tucker (KKT) optimality conditions are developed as a natural consequence of strong duality, encompassing primal feasibility, dual feasibility, complementary slackness, and the zero gradient condition. These conditions are shown to be both necessary and sufficient for optimality in convex problems, generalizing the simple unconstrained optimality condition ∇f(x) = 0. The lecture demonstrates these concepts through a detailed analysis of SVM, showing how KKT conditions lead to the identification of support vectors and the final classification boundary.

The presentation concludes by emphasizing the practical significance of duality in providing optimality certificates during algorithm execution, enabling real-time assessment of solution quality through the duality gap. This theoretical framework not only illuminates the mathematical structure underlying many machine learning algorithms but also provides the foundation for understanding why certain AI and optimization problems can be solved reliably while others remain computationally intractable. The lecture successfully bridges the gap between mathematical theory and practical applications in modern AI systems.