Correspondence: Gautam Mitra, CARISMA — The Centre for the Analysis of Risk Optimisation Modelling Applications, Brunel University, London, UK. Tel: +44 0 1895 265186; Fax: +44 0 1895 269732; E-mail: gautam.mitra@brunel.ac.uk

¹is an internationally renowned research scientist in the field of operational research in general and computational optimisation and modelling in particular. He qualified as an Electrical Engineer from Jadavpur University, India and obtained a PhD from London University in computer methods in operational research. He has developed a world-class research group in his area of specialisation, with researchers from Europe, the UK and the USA. He has published three books and over 100 refereed research articles. He was Head of the Department of Mathematical Science at Brunei University between August 1990 and July 2001 and is currently director of the Centre for the Analysis of Risk and Optimization Modelling Applications: CARISMA. He is also a Director of UNICOM Consultants (trading as OptiRisk Systems Ltd). Many of the research results are exploited through this company.

²was educated at Clifton College Bristol and at The Queen's College Oxford, where he received a first class honours degree in mathematics. Subsequently, he worked for several computer software companies and did work on mathematical programming, database and survey research systems. He was closely associated with the development of the Ophelie LP system, he designed and built the integer sub-system of APEX2 for CDC 6600 and 7600 mainframe computers, and he prepared the design for much of LP29Q0 for ICL 2900 series computers. His recent work with Brunei and OptiRisk Systems includes development and maintenance of the FortMP system, assistance with various post-graduate studies, and he has recently been awarded a PhD for his thesis on Quadratic Programming.

³was educated at Ruskin College, Oxford and Wolfson College, Cambridge, where he was awarded the Jennings Prize for academic achievement. He taught econometrics at Clare College, Cambridge before joining Phillips & Drew as an econometrician in 1984. There he worked with the leading macro research group at UBS and since that time he has worked on every aspect of quantitative modelling from stock selection to asset allocation. He has been closely associated with pioneering work on equity style and portfolio analysis developed by UBS Warburg. His research interests include practical applications of Bayesian econometrics and portfolio optimisation. Until his recent retirement in June 2006 he was the global Head of the top ranked equities quantitative research team at UBS Warburg.

Received 2 August 2007; Revised 2 August 2007.

Abstract

Convex quadratic programming as applied to portfolio planning is established and well understood. In this paper, presented in two parts, we highlight the importance of choosing an algorithm that processes a family of problems efficiently. In Part I (published in issue 8/3), in particular, we described an adaptation of the simplex method for Quadratic Programming (QP). The method not only takes advantage of the sparse features of simplex, the use of the duality property makes it ideally suited for processing the discrete optimisation models. Part II of the paper considers a family of discrete QP formulations of the portfolio problem, which capture threshold constraints and cardinality restrictions. We describe the adaptation a novel method 'branch, fix and relax' to process this class of models efficiently. Theory and computational results are presented.