Continuous Optimization, 7.5 Credits

Contents

Element 1 (4.5 hp) Theory
This part contains theory for nonlinear optimization. The course starts with a discussion of basic notions like classification of optimization problems, objective functions, constraints, feasible solutions, optimal solutions. Then fundamental convexity theory is addressed. A general optimization algorithm is defined, and notions like convergence rate, line search, descent- and ascent directions and optimality conditions are discussed. Optimality conditions for free optimization problems are introduced, and Newton's method for free optimization problems is studied. Next, optimization problems with constraints, Lagrange functions and Lagrange multipliers are defined, and optimility conditions (The Karush-Kuhn-Tucker conditions) are introduced. The duality concept and weak and strong duality theorems are discussed. Finally, some mathematical modeling examples leading to optimization problems are discussed.

Element 2 (3 hp) Computer labs
This part contains implementation of some approximation methods for free optimization problems (steepest descent, quasi-Newton method), and constrained optimization problmes (feasible-point methods, the SQP method). Furthermore, one assignment includes both mathematical modeling, formulation of an optimization model and finding an approximate solution of the model with suitable software.

Expected learning outcomes

For a passing grade, the student must be able to
 
Knowledge and understanding 

• explain fundamental concepts concerning the theory of convexity and duality

Skills and abilities

• formulate nonlinear optimization problems from given problem situations, and give necessary and sufficient conditions for optimal solutions.
• analyse and solve simple problems analytically.
• implement approximation methods for local optimal solutions and use them to solve more demanding optimization problems.
• use software for solving demanding global optimization problems.

Judgment and approach

• evaluate plausibility and efficiency for different optimization methods.

Required Knowledge

The course requires 60 ECTS in Mathematics and Mathematical Statistics or minimum 120 ECTS and in both cases courses in Multivariable Calculus and Differential Equations minimum 7,5 ECTS and a basic course in computer programming or equivalent.Proficiency in English equivalent to the level required for basic eligibility for higher studies. Where the language of instruction is Swedish, applicants must prove proficiency in Swedish to the level required for basic eligibility for higher studies.

Form of instruction

The teaching in element 1 takes the form of lectures and lessons. The teaching in element 2 takes the form of lab work and introductory lectures.

Examination modes

Element 1 is assessed through a written examination.. Element 2 is assessed through written lab reports. For element 1, one of the following evaluation is awarded: Fail (U), Pass (3), Pass with merit (4), Pass with distinction (5). For element 2, one of the following evaluation is awarded: Fail (U), or Pass (G).  For the course as whole, one of the following grades is awarded: Fail (U), Pass (3), Pass with merit (4), Pass with distinction (5). The grade for the whole course is determined by the grade given for element 1. To pass the whole course, all elements must have been passed. The grade is only set once all compulsory elements have been assessed.

A student who has been awarded a passing grade for the course cannot be reassessed for a higher grade. Students who do not pass a test or examination on the original date are given another date to retake the examination. A student who has sat two examinations for a course or a part of a course, without passing either examination, has the right to have another examiner appointed, provided there are no specific reasons for not doing so (Chapter 6, Section 22, HEO). The request for a new examiner is made to the Head of the Department of Mathematics and Mathematical Statistics. Examinations based on this course syllabus are guaranteed to be offered for two years after the date of the student's first registration for the course.
 
Credit transfers
Students are entitled to an assessment of whether previous education or equivalent knowledge and skills acquired in professional experience can be accredited for equivalent studies at Umeå University. Applications for credit transfers must be sent to Student Services/Degree Evaluation Office. More information on credit transfers can be found on Umeå University's student website, www.student.umu.se, and in the Higher Education Ordinance (Chapter 6). Rejected applications for credit transfers can be appealed (Higher Education Ordinance, Chapter 12) to the Higher Education Appeals Board. This applies regardless of whether the rejection relates to all or parts of the credit transfer application.

Other regulations

In a degree, this course may not be included together with another course with a similar content, such as Optimization 2 (5MA138). If unsure, students should ask the Director of Studies in Mathematics and Mathematical Statistics