EPSRC Reference: 
EP/P019676/1 
Title: 
Tropical Optimisation 
Principal Investigator: 
Sergeev, Dr S 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
School of Mathematics 
Organisation: 
University of Birmingham 
Scheme: 
First Grant  Revised 2009 
Starts: 
01 April 2017 
Ends: 
31 August 2019 
Value (£): 
101,183

EPSRC Research Topic Classifications: 
Mathematical Aspects of OR 


EPSRC Industrial Sector Classifications: 
Transport Systems and Vehicles 


Related Grants: 

Panel History: 

Summary on Grant Application Form 
Tropical mathematics is a relatively new branch of mathematics (initiated in the 1960's) where the usual set of arithmetical operations (+,*) is replaced with the tropical arithmetics in which the role of addition is played by the operation of taking maximum or minimum of two numbers. This appears to be useful in a number of problems which are nonlinear in the sense of "traditional" mathematics, but can be seen as linear over the tropical arithmetics. In particular, the whole Dutch railway network has been recently modelled as a huge tropical linear system, where the tools of tropical algebra can be exploited in order to evaluate viability and stability of the network and to visualise the propagation of delays. On the theoretical side, the development of tropical mathematics has been driven by search for analogies of useful facts, concepts and constructions of "traditional" mathematics.
The theme of the project is tropical optimisation. In principle, optimisation is one of the most applicable branches of mathematics. An ordinary person is solving various optimisation problems every day, for example, to minimise time and money that they spend or to maximise the impact of their effort. Mathematical formulation of an optimisation problem typically consists of an objective function to be optimised and a mathematical description of the area and variables over which the optimisation is performed. The project is focused, more specifically, on the case when both the function to be optimised and the optimisation domain can be most conveniently described in terms of tropical algebra.
We will solve problems of optimal control with switching tropical linear systems. In terms of railway scheduling applications that will mean optimising dispatchers' decisions in order to minimise the cumulative or maximal delays of train departures, thus responding to the public demand to increase the punctuality of railway operations. Since our approach will be rooted in tropical algebraic theory, new solutions to tropical optimal control problems will create new synergies between railway engineering and tropical mathematics communities.
We will also solve tropical pseudoquadratic programming and bilevel programming problems. Solutions of these problems will be further applied to project scheduling and urban planning problems with various synchronisation requirements. Let us point out here that bilevel optimisation has numerous applications in logistics and economics, since it handles real life situations which can be modelled as games between the leader and several followers who adjust their actions depending on the leader's policies but whose decisions might also influence leader's income. Solving the tropical analogues of these problems will enable us to deal with such situations also in the areas where the tropical optimisation is applied.

Key Findings 
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Potential use in nonacademic contexts 
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Impacts 
Description 
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Summary 

Date Materialised 


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Project URL: 

Further Information: 

Organisation Website: 
http://www.bham.ac.uk 