## Numerical Methods for Optimization and Optimal Control

(3-0-0-3)

**CMPE Degree:** This course is Elective for the CMPE degree.

**EE Degree:** This course is Elective for the EE degree.

**Lab Hours:** 0 supervised lab hours and 0 unsupervised lab hours.

**Technical Interest Group(s) / Course Type(s):** Systems and Controls

**Course Coordinator:** Anthony Joseph Yezzi

**Prerequisites:** ECE 3084 Signals and Systems OR ECE 3550 Feedback Control Systems

**Corequisites:** None.

### Catalog Description

Algorithms for numerical optimization and optimal control, Gradient-descent techniques, linear programming, numerical linear system solvers, second-order methods for optimizing performance of dynamical systems.### Textbook(s)

### Course Outcomes

- Students will be able to write computer code to implement linear system constructs learned in prior S+C courses.
- Students will understand the types of errors than can arise in these implementations, and how to quantify them.
- Students will be able to construct computer algorithms to solve other problems which don't benefit from closed form constructs acquired in previous
- Students will know which types of models should be paired with which types of numerical optimization algorithms based on properties such as convexi
- In cases were multiple choices of numerical optimization algorithms are suitable for a given problem, students will understand the relative advanta

### Student Outcomes

In the parentheses for each Student Outcome:"P" for primary indicates the outcome is a major focus of the entire course.

“M” for moderate indicates the outcome is the focus of at least one component of the course, but not majority of course material.

“LN” for “little to none” indicates that the course does not contribute significantly to this outcome.

1. ( P ) An ability to identify, formulate, and solve complex engineering problems by applying principles of engineering, science, and mathematics

2. ( LN ) An ability to apply engineering design to produce solutions that meet specified needs with consideration of public health, safety, and welfare, as well as global, cultural, social, environmental, and economic factors

3. ( LN ) An ability to communicate effectively with a range of audiences

4. ( LN ) An ability to recognize ethical and professional responsibilities in engineering situations and make informed judgments, which must consider the impact of engineering solutions in global, economic, environmental, and societal contexts

5. ( M ) An ability to function effectively on a team whose members together provide leadership, create a collaborative and inclusive environment, establish goals, plan tasks, and meet objectives

6. ( P ) An ability to develop and conduct appropriate experimentation, analyze and interpret data, and use engineering judgment to draw conclusions

7. ( M ) An ability to acquire and apply new knowledge as needed, using appropriate learning strategies.

### Strategic Performance Indicators (SPIs)

Not Applicable

### Course Objectives

- To teach students the elements of numerical and computational methods for optimization and optimal control.
- Students will become familiar with static optimization methods such as gradient descent and Newton’s method, and with optimization of dynamical system

### Topical Outline

I. Numerical Methods

I.1 Review of numerical issues when using machine number systems

I.2 Direct Linear systems solvers (LU, Cholesky, iterative refinement)

I.3 Scalar nonlinear equation solvers (bisection, Newton, secant method)

I.4 Numerical differentiation and Integration

II. Optimization

II.1. Introduction to nonlinear programming

II.2. Optimality conditions for constrained and unconstrained problems

II.3. Algorithms: gradient descent, Newton-Rhapson, conjugate-gradient techniques

II.4. Least-square problems and algorithms

II.5. Optimization problems with equality and inequality constraints

III. Algorithms for optimal control problems

III.1. Discrete-time optimal control: necessary optimality conditions

III.2. Continuous-time optimal control: the Hamiltonian

III.3. The linear-quadratic optimal control problem and feedback-based solution

III.4. The Pontryagin’s Maximum principle

III.5. Examples: minimum-time problems and minimal-path problems