Optimization

The main focus of the research activity of the group is on mathematical theory and methods applicable to managerial decision-making, particularly non-linear, discrete, and multi-criteria optimization. Research is typically a mixture of theoretical investigation and practical application or industrial collaboration.

 Bridge showing result of topology optimization starting from a dense mesh of potential bars

Image: bridge showing the rest of topology optimization starting from a dense very mesh of potential bars

Optimization Analysis interacts closely with Numerical Analysis, for example organising the biannual IMA Conference on Numerical Linear Algebra and Optimization since 2007 and the joint Optimization and Numerical Analysis seminar.

The research interests of the members of our group are listed below.

Professor M Kocvara

Professor M Kocvara is a Professor in Applied Mathematics

Algorithms for large-scale nonlinear and semidefinite optimization 
We are interested in the development of algorithms and software for large-scale optimization problems. We aim at combining efficient algorithms of mathematical optimization with powerful tools of numerical linear algebra. Particular interest is given to linear and nonlinear semidefinite programming problems, and problems with special data structures. Professor Kocvara is a co-author of a computer program PENNON which is the first known code that can solve optimization problems with a combination of standard non-linear and matrix inequality constraints. Possibilities for PhD work include, among others, development of novel algorithms using special structure of the underlying models, special data structure of the problems, and strong links to modern techniques of numerical linear algebra, like multigrid and domain decomposition.

Optimization of elastic structures
Our goal is to design optimal material properties and distribution within an elastic body. Emphasis is given to so-called Free Material Design which deals with the question of finding the lightest structure subject to one or more given loads when both the distribution of material and the material itself can be freely varied. The techniques and tools of material and topology optimization accelerated rapidly in the last ten years and recently found a way to many industrial companies, in particular in the automotive and aircraft industry. Prof. Kocvara is a member of the team that developed a computer code MOPED used, among others, in the design of components of the new Airbus A380. Possibilities for PhD work include, among others, development of new mathematical models for topology and material optimization based, in particular, on (nonlinear) semidefinite programming.

Professor M Kocvara profile

Dr Sandor Nemeth

Senior Lecturer

Cones and ordering
The Cone is an important mathematical object which occurs in many application oriented investigations. The order structure of a linear space is determined by its positive cone. Several new results were obtained in this topic. Some of the results were used for complementarity problems and finding the projection onto conesof special structure, but many results have not been applied yet. S. Z. Németh has introduced the notion of is otone retraction cones which gives a nice connection between the geometry, topology and ordering structure of the space.

Projection onto polyhedral cones and applications  
Projecting efficiently onto high dimensional polyhedral cones is an open problem with many practical and theoretical applications. Possible applications are regression, image reconstruction, pattern recognition, complementarity problems, non-negative solutions of linear systems of equations etc. The particular problem of projecting efficiently onto simplicial cones is also open and important. Recently A. Ekárt, A. B. Németh & S. Z. Németh have developed a seemingly efficient heuristic method for projecting onto simplicial cones. At the moment the method is empirical only and there is need to develop a theoretical foundation for it. For isotone projection cones (simplicial cones with a special ordering structure) A. B. Németh & S. Z. Németh have developed an efficient method of projection. Any new result in this topic is very important and can open new opportunities for applications. A. B. Németh & S. Z. Németh are preparing a book about Euclidean Vector Lattices which deals with the problem of projecting onto simplicial cones too.

Dr Sandor Nemeth profile

Dr Sergey Sergeev

Lecturer in Mathematical Optimisation

Since March 2014 Sergey has been a Lecturer in Mathematical Optimisation. Currently he is working on an EPSRC funded grant EP/P019676/1 "Tropical Optimisation", being focused on various forms of tropical optimisation such as tropical pseudoquadratic and bilevel optimisation. He has been active as organiser of one-day minisymposia in Tropical Mathematics such as a minisymposium on Tropical Mathematics and its Applications in the framework of ILAS-2016 conference in Leuven, Belgium and LMS Workshop on Tropical Mathematics held in Birmingham, November 2017. He is teaching Non-Linear Programming II, a module for 4th year students and Ph.D students which is mostly on Dynamic Programming and Optimal Control.

Dr Sergey Sergeev profile

Dr Daniel Jones

Lecturer in  Game Theory and Combinatorial Optimisation

Dr Daniel Jones profile

Weekly seminar

We host a weekly seminar with invited external speakers. Please visit our Seminars page for details.

PhD and MPhil research opportunities

Research students are welcomed in to  the research group and are actively involved in all research activities.  To find out more about PhD and MPhil opportunities please see the Graduate School for Mathematics.