Research interests
My research is on the theory and implementation of Interior Point Methods for linear and quadratic programming. My interests lie particularly in the study of search directions and warm-start approaches.
During my PhD I have improved the implementation of corrector directions in the HOPDM interior point solver (MS 05-001), before moving to work with the structure-exploiting parallel solver OOPS. I have implemented an SMPS interface for OOPS, HOPDM and CPLEX (now extended to LP_SOLVE and GLPK), which allows to solve a stochastic programming problem by formulating the corresponding deterministic equivalent problem. This implementation allows to solve a problem instance with warm-start, by first solving a problem with reduced dimensions (MS 06-004).
I have been employed on an EPSRC funded project with Andreas Grothey, in which we continued the investigation of warm-start strategies for interior point methods in the context of stochastic programming. This led to consider a multi-step approach, in which the number of intermediate problems can be more than one (ERGO 09-007), and a decomposition-like strategy, in which we generate and warm-start the subproblems rooted at the second-stage nodes (ERGO 09-008). I also contributed the stochastic programming extension for the structure-conveying modelling language SML (ERGO 09-003).
Since October 2009 I've been a post-doctoral research assistant at the Centre for Population Health Sciences, working on the genetic epidemiology software admixmap and on applications of machine learning algorithms to biomarker screening and prediction from high-dimensional data.
Research papers and technical reports
P.M. McKeigue, M. Colombo, F. Agakov, I. Datta, A. Levin, D. Favro, R.R. Burke, C. Gray-Montgomery, M.C. Iannuzzi and B.A. Rybicki Extending admixture mapping to nuclear pedigrees: application to sarcoidosis To appear in Genetic Epidemiology.
B.A. Rybicki, A.M. Levin, P. McKeigue, I. Datta, C. Gray-McGuire, M. Colombo, D. Reich, R.R. Burke and M.C. Iannuzzi A genome-wide admixture scan for ancestry-linked genes predisposing to sarcoidosis in African-Americans Genes and Immunity, 12, 67-77, 2011.
M. Colombo and A. Grothey
A decomposition-based warm-start method for stochastic programming
Technical Report ERGO 09-008
School of Mathematics, University of Edinburgh
To appear in Computational Optimization and applications.
M. Colombo and A. Grothey
A multi-step interior point warm-start approach for large-scale stochastic linear programming
Technical Report ERGO 09-007
School of Mathematics, University of Edinburgh
M. Colombo, A. Grothey, J. Hogg, K. Woodsend and J. Gondzio
A structure-conveying modelling language for mathematical and stochastic programming
Technical Report ERGO 09-003
School of Mathematics, University of Edinburgh
Mathematical Programming Computation, 1 (4), 223-247, 2009.
A. Grothey, J. Hogg, K. Woodsend, M. Colombo and J. Gondzio A structure-conveying parallelisable modelling language for mathematical programming "Parallel Scientific Computing and Optimization: Advances and Applications", R. Ciegis, D. Henty, B. Kågström, J. Zilinskas (eds.), Springer, 2009.
M. Colombo, J. Gondzio and A. Grothey
A warm-start approach for large-scale stochastic linear programs
Technical Report MS 06-004
School of Mathematics, University of Edinburgh
Mathematical Programming, 127 (2), 371-397, 2011.
M. Colombo and J. Gondzio
Further development of multiple centrality correctors for interior point methods
Technical Report MS 05-001
School of Mathematics, University of Edinburgh
Computational Optimization and Applications, 41 (3), 277-305, 2008.
Dissertations
M. Colombo
Advances in interior point methods for large-scale linear programming
Thesis presented for the Degree of PhD in Optimization
School of Mathematics, University of Edinburgh
Successfully defended on 6 September 2007, examiners Danny Ralph and Ken McKinnon.
M. Colombo
A branch-and-cut approach to solve the Hamiltonian Cycle Problem
Dissertation presented for the Degree of MSc in Operational Research
School of Mathematics, University of Edinburgh
MSc in OR awarded with distinction.
Software
I have been developing a variety of software related to interior point methods and stochastic programming. In due course, I expect them to be ready for public consumption, and appropriate links will appear down here. In the meantime, if you are interested in any of these, get in touch!
libsmps++
A C++ interface to read and solve problems in SMPS format.
smps-script
A python library to help the generation of stochastic problems in SMPS format.
hippy
An implementation of interior point methods in python.
You may also want to check out other interesting software being worked on by the ERGO research group, in particular OOPS (Object-Oriented Parallel Solver) and SML (Structured Modelling Language).
Presentations
M. Colombo A decomposition-based warm-start method for stochastic programming ERGO Seminar, School of Mathematics, University of Edinburgh
M. Colombo OOPS: a structure-exploiting parallel solver Cariplo workshop on numerical stochastic programming, Edinburgh
M. Colombo, A. Grothey Stage aggregation to warm-start interior point methods APMOD 2008, Bratislava
M. Colombo OR methods in channel coding and code design Mathematical Techniques in Coding Theory workshop, Edinburgh
M. Colombo, J. Gondzio, A. Grothey A warm-start approach for large-scale stochastic linear programs SPXI, Vienna
M. Colombo, J. Gondzio, A. Grothey Solution techniques for large-scale financial planning problems ICCOPT II - MOPTA 2007, Hamilton
M. Colombo, J. Gondzio, A. Grothey A warm-start approach for large-scale stochastic linear programs Spring School on Stochastic Programming, Bergamo
M. Colombo, J. Gondzio Further developments of multiple centrality correctors Euro 2006, Reykjavik
M. Colombo, J. Gondzio, A. Grothey A new strategy to start interior point methods for stochastic linear programming problems Apmod 2006, Madrid
M. Colombo, J. Gondzio A computational experience with search directions in interior point methods Dundee NA 2005, Dundee
M. Colombo A branch-and-cut approach to solve the Hamiltonian Cycle Problem ERGO Seminar, School of Mathematics, University of Edinburgh