Convex Optimization

Course Description

The course concentrates on recognizing and solving convex optimization problems that arise in applications. Basics of linear programming. Convex sets, convex functions, and convex optimization problems. Basics of convex analysis. Least-squares, linear and quadratic programs, linear matrix inequalities, semidefinite programming. Optimality conditions, duality theory, theorems of alternative, and applications. Interior-point methods. Applications to signal processing, statistics and machine learning, control and mechanical engineering, digital and analog circuit design, and finance.

Learning Outcomes

  1. define a linear programming problem and identify the associated dual problem
  2. explain the idea of the simplex method
  3. breakdown the steps of the primary-dual inner-point method for linear problems
  4. identify convex optimization problem
  5. analyze Karush-Kuhn-Tucker optimality conditions
  6. create a dual problem and analyze the sensitivity of the optimum to the perturbations of the constraints
  7. recognize the problem of semidefinite programming
  8. apply and modify appropriate numerical conditional optimization methods

Forms of Teaching

Lectures

3 hours of lectures per week

Grading Method

Continuous Assessment Exam
Type Threshold Percent of Grade Threshold Percent of Grade
Class participation 0 % 5 % 0 % 5 %
Mid Term Exam: Written 0 % 20 % 0 %
Final Exam: Written 0 % 25 %
Final Exam: Oral 50 %
Exam: Written 0 % 45 %
Exam: Oral 50 %

Week by Week Schedule

  1. Convexity; Polyhedra, convex hulls, polytopes
  2. The simplex method
  3. Duality and sensitivity, Primal-dual interior-point method for linear programming
  4. Convex sets and convex functions
  5. Convex optimization problems
  6. Quadratic problems (box-constrained, equality constrained, inequality constrained)
  7. Tangent cone and constraint qualifications
  8. Midterm exam
  9. Karush-Kuhn-Tucker optimality conditions
  10. Duality; The Lagrange function; Perturbation and sensitivity analysis
  11. Linear matrix inequalities; Eigenvalue problems
  12. Semidefinite programming
  13. Active set methods for convex quadratic problems
  14. Interior point methods for convex optimization problems
  15. Final exam

Study Programmes

University graduate
Audio Technologies and Electroacoustics (profile)
Free Elective Courses (1. semester) (3. semester)
Communication and Space Technologies (profile)
Free Elective Courses (1. semester) (3. semester)
Computational Modelling in Engineering (profile)
Free Elective Courses (1. semester) (3. semester)
Computer Engineering (profile)
Free Elective Courses (1. semester) (3. semester)
Computer Science (profile)
Free Elective Courses (1. semester) (3. semester)
Control Systems and Robotics (profile)
Elective Courses of the Profile (1. semester)
Data Science (profile)
Free Elective Courses (1. semester) (3. semester)
Electrical Power Engineering (profile)
Free Elective Courses (1. semester) (3. semester)
Electric Machines, Drives and Automation (profile)
Free Elective Courses (1. semester) (3. semester)
Electronic and Computer Engineering (profile)
Free Elective Courses (1. semester) (3. semester)
Electronics (profile)
Free Elective Courses (1. semester) (3. semester)
Information and Communication Engineering (profile)
Free Elective Courses (1. semester) (3. semester)
Network Science (profile)
Free Elective Courses (1. semester) (3. semester)
Software Engineering and Information Systems (profile)
Free Elective Courses (1. semester) (3. semester)

Literature

Stephen Boyd, Stephen P. Boyd, Lieven Vandenberghe (2004.), Convex Optimization, Cambridge University Press
Jorge Nocedal, Stephen Wright (2006.), Numerical Optimization, Springer Science & Business Media
Igor Griva, Stephen G. Nash, Ariela Sofer (2009.), Linear and Nonlinear Optimization, SIAM

For students

General

ID 222633
  Winter semester
5 ECTS
L1 English Level
L1 e-Learning
45 Lectures

Grading System

87.5 Excellent
75 Very Good
62.5 Good
50 Acceptable