1

Introduction and Overview of the Course

2

Overview of Nonlinear and Convex Optimization:

i) Optimality Conditions

ii) Newton’s Method

iii) Interior Point Method

iv) Dual Optimization

v) Convex Optimization

vi) Linear Program

vii) Semidefinite Program

Problem set 1 out

3

Nonlinear Optimization Using the Theory of Nonnegative Polynomials

Sum-of-Squares Formulation (SOS)

Problem set 1 due

Problem set 2 out

4

Nonlinear Optimization Using the Theory of Measure and Moments

Problem set 2 due

Problem set 3 out

5

Duality

i) Duality of Moments and Polynomials

ii) Duality of Measures and Continuous Functions

Problem set 3 due

Problem set 4 out

6

Modified Sum-of-Squares Optimization

i) Spars Sum-of-Squares Optimization (SOS)

ii) Bounded Degree Sum-of-Squares Optimization (BSOS)

iii) (Scaled) Diagonally Dominant Sum-of-Squares Optimization (SDSOS, DSOS)

Problem set 4 due

Problem set 5 out

7

Chance Optimization and Chance-Constrained Optimization

i) Measure and Moments Formulation

ii) Sum-of-Squares Formulation

Problem set 5 due

8

Part 1: Robust Optimization Using Sum-of-Squares Optimization

Part 2: Distributionally Robust Chance-Constrained Optimization

Problem set 6 out

9

Algorithms for Large-Scale Semidefinite Programs (Guest Lecture)

Problem set 6 due 

Problem set 7 out

10

Safety Verification of Probabilistic Systems

i) Risk Estimation

ii) Probabilistic Uncertainty Propagation

iii) Uncertainty Set Construction

iv) Forward-Reachable Sets

Problem set 7 due

11

Risk Aware Planning and Control

i) Risk-Bounded Trajectory Planning

ii) Risk Aware Nonlinear Control

iii) Flow-Tube–Based Control

iv) Backward-Reachable Sets

12

Dynamical Systems with Gaussian Uncertainties

i) Chance-Constrained Control

ii) Safety Verification

iii) Distributionally Robust Chance Constraints

13

Occupation Measure-Based Analysis and Control

i) Safety Verification

ii) Region-of-Attraction Set

iii) Optimal Control

14

Sum-of-Squares Optimization for Uncertain Nonlinear Systems

i) Lyapunov-Based Stability and Region-of-Attraction Set

ii) Barrier-Function–Based Safety Verification

iii) Robust Control

15

Final Project Presentation