ECE 490 – weekly plan

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

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