Lectures in course MATH-329: Continuous optimization

Introduces linear algebra, calculus, and optimization basics in Euclidean spaces, emphasizing the power of optimization as a modeling tool.

Covers the Lipschitz gradient theorem and its applications in function optimization.

Explores Lipschitz continuity in gradient descent optimization and its implications on function optimization.

Explores convex functions, global minima, and their relationship with differentiability.

Explores singular value decomposition and its role in understanding linear maps.

Explores the convergence of Newton's method and the CG algorithm for solving linear equations.

Covers the Hessian and Newton's method for optimization using iterative processes.

Explores the analysis of trust regions with Cauchy steps in optimization.

Covers trust region methods, focusing on the framework and algorithms.

Covers the truncated conjugate gradients algorithm for solving positive definite linear maps iteratively.

Covers the trust-region method and introduces the truncated conjugate gradients method.

Covers the basics of constrained optimization, including tangent directions, trust-region subproblems, and necessary optimality conditions.

Covers necessary optimality conditions for optimization with constraints and discusses cones and polar sets.

Explains feasible directions, linearized cone, and Constraint Qualification conditions in optimization.

Covers single inequality or equality constraints and necessary optimality conditions in optimization problems.

Covers the KKT conditions and convex optimization, discussing constraint qualifications and tangent cones of convex sets.

Covers the concept of the polar of the cone of linearized directions and its special cases.

Covers the KKT conditions and Slater's condition in convex optimization problems.

Explores optimization on convex sets, including KKT points and tangent cones.

Covers the proof of strong duality in optimization problems and provides examples of Rayleigh quotient optimization.