week of Jan 14

**Course Overview**

**Differential Calculus**

Taylor’s Formula.

Differentiation of Functions of Several Variables.

Differentiability of Vector-Valued Functions.

Jacobians.

Chain Rule.

Taylor’s Formula for Functions of Several Variables.

week of Jan 21

**Unconstrained Optimization**

Basic Results on the Existence of Optimizers.

First-Order Optimality Conditions.

Second-Order Optimality Conditions

Quadratic Forms.

week of Jan 28

**Global Analysis Tools
**

The Inverse Function

Implicit Function

Lyusternik Theorems in Finite Dimensions.

Morse’s Lemma.

Ekeland’s variational principle.

Gordan’s lemma.

week of Feb 4

**Convex Analysis
**

Affine Geometry.

Convex Sets.

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

**Linear Programming**

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

Simplex Method.

**Nonlinear Programming**

First-Order Necessary Conditions (Fritz John Optimality Conditions

First-Order Sufficient Optimality Conditions

week of Mar 4

Constraint Qualifications

Examples of Nonlinear Programs

Second-Order Conditions in Nonlinear Programming

**Duality Theory and Convex Programming.**

Saddle Points and Their Properties.

week of Mar 11

**Midterm**

Nonlinear Programming Duality.

week of Mar 18

**No classes**

week of Mar 25

Strong Duality in Convex Programming.

Examples of Dual Problems.

Quadratic Programming.

week of Apr 1

Conic Programming Duality.

Semi-definite programs.

**Basic Optimization Algorithms**

*Gradient-Descent Methods.*

Gradient Projection Method

week of Apr 8

*Newton’s Method*

Convergence Theory of Kantorovich

week of Apr 15

*Conjugate-Gradient Method*

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

**Semi-infinite programming.**

Fritz John Conditions for Semi-infinite Programming

Jung’s Inequality

Applications.

**Applications**

Supporting Vector Machines

LASSO

week of Apr 29

**Projects presentations**

**Course recap**