# Courses

## Convex Optimisaton, 2018 courses dates to be confirmed

PRE-REQUISITES:

1. Undergraduate level knowledge of linear algebra (e.g. the relevant chapter in Winston’s textbook on OR) and calculus (e.g. basic notions of continuity and differentiability).

2. The students are expected to have familiarized themselves with the material marked with an asterisk in the teaching schedule and with other directed reading.

AIMS OF THE COURSE

1. To develop knowledge of different aspects of convex optimization and its applications.

2. To develop an ability to model real life problems as mathematical programming problems and an ability to adapt industry standard solvers to process them.

3. To develop an ability to analyze optimization algorithms for their merits and shortcomings.

4. To develop an ability to work independently as well as in a peer group with limited supervision.

LEARNING OUTCOMES FOR THE COURSE

The course provides opportunities for students to develop and demonstrate knowledge and understanding, qualities, skills and other attributes in the following areas:

(A) Knowledge and Understanding

On successful completion of this course, the students will have

1. knowledge of theoretical underpinning of convexity in optimization and of general nonlinear programming methods. This knowledge will act as a foundation to understand an advanced graduate textbook or a research paper without significant help.

2. understanding of linear programming methods and related theoretical issues.

3. knowledge of semi-definite programming.

4. ability to use an industry standard optimization software system for processing optimization models.

(B) Cognitive Skills

On successful completion of this course, the students will be able to

5. formulate realistic industrial problems as mathematical programming problems.

6. analyze critically the choice of algorithms for solving different classes of a particular optimization model regarding their computational effectiveness.

7. construct elementary proofs related to the properties of optimization methods.

(C) Other Skills and Attributes (Practical/Professional/Transferable)

On successful completion of this course, the students will be able to

8. plan and execute a solution to an optimization problem as a group and will be able to present the results to peers and tutors.

PRINCIPAL TOPICS OF STUDY:

Foundations of Convexity: affine and convex sets, convex functions, composition of convex functions.

General Convex Optimization: examples of convex optimization problems, duality, unconstrained minimization, steepest descent method, first and second order optimality conditions in unconstrained minimization, Newton's method and convergence analysis, norm approximation problems.

Linear Programming: simplex method, duality for LP, interior point methods for LP.

Convex Quadratic Programming: simplex method for quadratic programming, application in finance, KKT conditions for convex QP.

Semi-definite Programming: formulation, extension of interior point methods to SDP, quadratically constrained convex quadratic programs.

A case study of convex optimization in industry.

Numerical Linear Algebra: algorithm complexity, Cholesky factorization, sparsity.

The latter part of this course will be run in two parallel streams: an application stream (A stream) and a theory stream (T stream). The lectures and workshops for the two streams will differ for the two streams for a part of the course, and the students will need to choose beforehand which stream they prefer to follow. Most of the topics of study are the same for both the streams, apart from the topics mentioned below:

A stream: Use of Industry Strength Solver Systems for LP/QP to process industrial problems.

T stream: advanced topics in optimization including interior point methods for convex quadratic optimisation, complexity analysis via self-concordance, interior point methods for second order cone programming, Nesterov’s method for smooth and non-smooth programming.

REFERENCES

[1] S. Boyd and L. Vandenberghe, *Convex Optimization*, Cambridge University Press, 2004. As of June 2009, this text is freely available to download at http://www.stanford.edu/~boyd/cvxbook/ . The chapters relevant to this course are 2-5 and 9-11.

[2] D.G. Luenberger, *Linear and Nonlinear Programming*, Kluwer, 2003. The chapters relevant to this course are 2-10.

[3] R. Fourer, .M. Gay, B.W. Kernighan, *Ampl: A Modeling Language for Mathematical Programming*, Brooks Cole, 2002.

[4] Hillier and Lieberman, *Introduction to Operations Research*, McGraw Hill, 2002. The chapter relevant to this course is 13.

**Start Date:**TBC**End Date:**TBC**Location:****Address:****Postcode:**