A Continuous Optimization Approach for the Financial Portfolio Selection under Discrete Asset Choice Constraints

In this paper we consider a generalization of the Markowitz’s Mean-Variance model under linear transaction costs and cardinality constraints. The cardinality constraints are used to limit the number of assets in the optimal portfolio. The generalized model is formulated as a mixed integer quadratic programming (MIP) problem. The purpose of this paper is to investigate a continuous approach based on difference of convex functions (DC) programming for solving the MIP model. The preliminary comparative results of the proposed approach versus CPLEX are presented.

💡 Research Summary

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The paper addresses a realistic extension of the classic Markowitz mean‑variance portfolio selection model by incorporating linear transaction costs and a cardinality constraint that limits the number of assets held in the final portfolio. These additions turn the originally convex quadratic program into a mixed‑integer quadratic programming (MIP) problem, which is NP‑hard and difficult to solve exactly for moderate to large numbers of assets.

To overcome this difficulty, the authors propose a continuous‑optimization approach based on Difference‑of‑Convex (DC) programming and the associated DC Algorithm (DCA). The key idea is to replace the hard cardinality restriction with an exact penalty term α(z)=θ∑_j z_j(1−z_j), where z_j∈{0,1} indicates whether asset j is selected. For a sufficiently large θ, α(z) is zero only when each z_j is binary, and positive otherwise, thereby penalising fractional selections. Adding this concave penalty to the original objective yields a DC objective: a convex part (the original quadratic risk term plus linear transaction‑cost terms and indicator of the feasible polyhedron) minus the concave penalty α(z).

The DC formulation enables the use of DCA, which iteratively linearises the concave component at the current point, solves a convex quadratic subproblem, and updates the variables. Specifically, at iteration k the subgradient of α at z^k is computed analytically (the expression is simple because α is separable), and a convex QP of the form

 min  xᵀQx + cᵀx + ( subgradient )ᵀz

subject to the linear constraints (budget, return target, bounds, and binary‑relaxed z∈