A new method for mean-variance portfolio optimization with cardinality constraints

Abstract

Several portfolio selection models take into account practical limitations on the number of assets to include and on their weights in the portfolio. We present here a study of the Limited Asset Markowitz (LAM) model, where the assets are limited with the introduction of quantity and cardinality constraints.\ud We propose a completely new approach for solving the LAM model based on a reformulation as a Standard Quadratic Program, on a new lower bound that we establish, and on other recent theoretical and computational results for such problem. These results lead to an exact algorithm for solving the LAM model for small size problems. For larger problems, such algorithm can be relaxed to an efficient and accurate heuristic procedure that is able to find the optimal or the best-known solutions for problems based on some standard financial data sets that are used by several other authors. We also test our method on five new data sets involving real-world capital market indices from major stock markets. We compare our results with those of CPLEX and with those obtained with very recent heuristic approaches in order to illustrate the effectiveness of our method in terms of solution quality and of computation time. All our data sets and results are publicly available for use by other researchers

The multiplicative weights update algorithm for mixed integer nonlinear programming: theory, applications, and limitations

“…Several authors (see, e.g., [44,49,97,133]) require x to be a semi-continuous variable [234], i.e., they require x j ∈ [x j , x j ] ∪ {0} for all j = 1, . .…”

“…Using the theoretical results in [224,236] and extending [45,46,47], Cesarone et al [44] have shown that the problem (3.1) with cardinality constraints (3.16) has the same optimal solution of problem (3.1) with equality cardinality constraints (3.17) and reduce this kind of programs to Standard Quadratic Programming Problem (see [23,24]), avoiding to explicitly introduce binary variables and considering an exact tailored solving procedure, called Increasing Set Algorithm. The Standard Quadratic Programming Problem is an NP-hard problem when the Hessian matrix of the objective function is indefinite, i.e., if the Hessian matrix of the objective function is neither positive nor negative semidefinite [23].…”

“…Finally, in several papers (see, e.g., [44,49,133]) it is observed that the problem (3.1), with cardinality constraints (3.16) and with minimum and maximum buy-in thresholds (3.10) can be straightforwardly reformulated as a convex MIQP.…”

A Survey on Financial Applications of Metaheuristics

“…For solving the problem, these authors propose a PSO algorithm. Cesarone et al [2013] analyze a cardinality-constrained version of the POSP. The authors develop an exact algorithm for solving small size instances of the problem.…”

Efficient Cardinality/Mean-Variance Portfolios

“…At a first glance, one could see the problem as a triobjective optimization problem by minimizing the variance of the return, maximizing the expected return, and minimizing the cardinality over the set of feasible portfolios. Such a framework was taken in account in the studies [1,2,10,18]. However, these authors did not investigate the effects of cardinality constraints on portfolio models in terms of out-of-sample performance, a subject still poorly analyzed in the literature.…”