Bounds on efficient outcomes for large-scale cardinality-constrained Markowitz problems
Due to the globalization of financial markets, investors have access to thousands of financial instruments (assets). The portfolio selection problem can be modelled with the use of the famous Markowitz's mean-variance (bi-objective) model -- the mean-variance problem (MVP) -- or its extensions. However, solving large-scale cardinality-constrained Markowitz mean-variance portfolio investment problems, exact solvers may be unable to derive some efficient portfolios, even within a reasonable time limit. In such cases, information on the distance from the best feasible solution, found before the optimization process has stopped, to the true efficient solution is unavailable. Moreover, portfolio management with a large number of assets might be costly, especially when an investor makes investment decisions relatively often. In order to limit the number of assets in efficient portfolios, a cardinality constraint can be added to the MVP. This leads to the cardinality-constrained Markowitz mean-variance portfolio investment problem (CCMP). The present paper focuses is the (bi-objective) CCMP. To start with, the author recalls the bi-objective problem formulation in its most general form and also recalls a method for the derivation of efficient solutions which is instrumental for the subsequent development. Then the CCMP is formulated and some of its properties that motivate to apply so called lower and upper bounds on values of objective functions of implicit efficient solutions, are considered. Taking this into account, the author shows how to calculate the lower and upper bounds on objective function values of implicitly given efficient solutions, mentioned above, and provides a method of populating lower and upper shells to the CCMP. Chebyshev scalarization is one of the main mathematical tools. Afterwards, the theory is illustrated by demonstration how the bounds work on a large-scale instance of the CCMP when a time limit on optimization is assumed.
cardinality-constrained Markowitz portfolio investment problem
two-sided efficient frontier approximations
scientific article; zbMATH DE number 7382628
scientific article