week of Jan 14
Differentiation of Functions of Several Variables.
Differentiability of Vector-Valued Functions.
Taylor’s Formula for Functions of Several Variables.
week of Jan 21
Basic Results on the Existence of Optimizers.
First-Order Optimality Conditions.
Second-Order Optimality Conditions
week of Jan 28
Global Analysis Tools
The Inverse Function
Lyusternik Theorems in Finite Dimensions.
Ekeland’s variational principle.
week of Feb 4
Convex Functions. Differentiable Convex Functions.
Optimization on Convex Sets.
Separation of Convex Sets.
Projection of a Point onto Closed Convex Set.
Separation of Convex Sets in Finite-Dimensional Vector Spaces.
Applications of Separation Theorems.
week of Feb 11
Proper Separation of a Convex Set and a Convex Polyhedron.
Dubovitskii–Milyutin Theorem in Finite Dimensions
Fundamental Theorems of Linear Programming
Dual Linear Program.
week of Feb 18
Duality Rules in Linear Programming.
Geometric Formulation of Linear Programs.
Strictly Complementary Optimal Solutions.
week of Feb 25
First-Order Necessary Conditions (Fritz John Optimality Conditions
First-Order Sufficient Optimality Conditions
week of Mar 4
Examples of Nonlinear Programs
Second-Order Conditions in Nonlinear Programming
Duality Theory and Convex Programming.
Saddle Points and Their Properties.
week of Mar 11
Nonlinear Programming Duality.
week of Mar 18
week of Mar 25
Strong Duality in Convex Programming.
Examples of Dual Problems.
week of Apr 1
Conic Programming Duality.
Basic Optimization Algorithms
Gradient Projection Method
week of Apr 8
Convergence Theory of Kantorovich
week of Apr 15
Convergence Rate of the Conjugate-Gradient Method.
The Conjugate-Gradient Method and Orthogonal Polynomials.
Convergence Rates for First Order Methods
Lower bounds for first order method.
Nesterov algorithm and its versions.
week of Apr 22
Fritz John Conditions for Semi-infinite Programming
Supporting Vector Machines
week of Apr 29