Convex relaxations and MIQCQP reformulations for a class of cardinality‐constrained portfolio selection problems

In this paper we investigate a class of cardinality‐constrained portfolio selection problems. We construct convex relaxations for this class of optimization problems via a new Lagrangian decomposition scheme. We show that the dual problem can be reduced to a second‐order cone program problem which is tighter than the continuous relaxation of the standard mixed integer quadratically constrained quadratic program (MIQCQP) reformulation. We then propose a new MIQCQP reformulation which is more efficient than the standard MIQCQP reformulation in terms of the tightness of the continuous relaxations. Computational results are reported to demonstrate the tightness of the SOCP relaxation and the effectiveness of the new MIQCQP reformulation.