Abstract
This paper deals with the issue of buy-in thresholds in portfolio optimization using the Markowitz approach. Optimal values of invested fractions calculated using, for instance, the classical minimum-risk problem can be unsatisfactory in practice because they lead to unrealistically small holdings of certain assets. Hence we may want to impose a discrete restriction on each invested fraction y i such as y i > y min or y i = 0. We shall describe an approach which uses a combination of local and global optimization to determine satisfactory solutions. The approach could also be applied to other discrete conditions—for instance when assets can only be purchased in units of a certain size (roundlots).
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References
Bartholomew-Biggs M, Parkhurst S, Wilson S(2003) Global optimization approaches to an aircraft routing problem. Eur J Oper Res 146:417–431
Bartholomew-Biggs M, Ulanowski ZJ, Zakovic S(2005) Using global optimization for a microparticle identification problem with noisy data. J Global Optim (32):325–347
Bixby R, Fenelon M, Gu Z, Rothberg E, Wunderling R(2000) MIP: theory and Practice—closing the gap. In: Powell MJD, Scholtes S (eds) System modelling and optimization methods theory and applications. Kluwer, Dordrecht
Broyden CG(1970a) The convergence of a class of double rank minimization algorithms. Part 1. J Inst Math Appl 6:76–90
Broyden CG(1970b) The convergence of a class of double rank minimization algorithms. Part 2. J Inst Math Appl 6:222–231
Ellison E, Hadjan M, Levkovitz R, Maros I, Mitra G, Sayers D (1999) FortMP Manual, OptiRisk Systems and Brunel University
Fletcher R, Leyffer S(1994) Solving mixed integer nonlinear programs by outer approximation. Math Program 66:327
Jobst NJ, Horniman MD, Lucas CA, Mitra G(2001) Computational aspects of alternative portfolio selection models in the presence of discrete asset choice constraints. Quant Finance 1:1–13
Jones DR, Perttunen CD, Stuckman BE (1993) Lipschitzian optimization without the Lipschitz constant. J Opt Theory Appl 79:157–181
Jones DR (2001) The DIRECT global optimization algorithm, In: Floudas CA, Pardolos PM (eds) Encyclopaedia of optimization. Kluwer, pp 431–440
Markowitz HM (1952) Portfolio selection. J Finance 7:77–91
Markowitz HM (1959) Portfolio selection: efficient diversification of investments, Wiley, New York (1959), 2nd edn. Blackwell, Oxford (1991)
Mitchell J, Borchers B (1997) A comparison of branch and bound and outer approximation methods for 0–1 MINLP. Comput Oper Res 24:699–701
Mitchell J, Braun S (2002) Rebalancing an investment portfolio in the presence of transaction costs, http://www.rpi.edu/ mitchj/papers/transcosts.html, Rensselaer Polytechnic Institute.
Mitra G, Kriakis T, Lucas C, Pirbhai M (2003) A review of portfolio planning: models and systems. In: Stachell SE, Scowcroft A (eds) Advances in portfolio construction and implementation. Butterworth-Heinemann, Oxford
Rinnooy Kan A, Timmer GT (1987a) Stochastic global optimization methods. Part I: clustering methods. Math Program 39:27
Rinnooy Kan A, Timmer GT(1987b) Stochastic global optimization methods. Part II: multilevel methods. Math Program 39:57
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Bartholomew-Biggs, M.C., Kane, S.J. A global optimization problem in portfolio selection. Comput Manag Sci 6, 329–345 (2009). https://doi.org/10.1007/s10287-006-0038-4
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DOI: https://doi.org/10.1007/s10287-006-0038-4