A sparse enhanced indexation model with norm and its alternating quadratic penalty method

Optimal investment strategies for enhanced indexation problems have attracted considerable attentions over the last decades in the field of fund management. In this paper, a featured difference from the existing literature is that our main concern of the investigation is the development of a sparse-enhanced indexation model to describe the process of assets selection by introducing a sparse ℓ1/2$ \ell _{1/2} $ regularization instead of binary variables, which is expected to avoid the over-fitting and promote a better out-of-sample performance for the resulting tracking portfolio to some extent. An Alternating Quadratic Penalty (AQP) method is proposed to solve the corresponding nonconvex optimisation problem, into which the Block Coordinate Descent (BCD) algorithm is integrated to solve a sequence of penalty subproblems. Under some suitable assumptions, we establish that any accumulation point of the sequence generated by the AQP method is a KKT point of the proposed model. Computational results on five typical data-sets are reported to verify the efficiency of the proposed AQP method, including the superiority of the sparse ℓ1/2$ \ell _{1/2} $ model with the AQP method over one cardinality constrained quadratic programming model with one of its solution methods in terms of computational costs, out-of-sample performances, and the consistency between in-sample and out-of-sample performances of the resulting tracking portfolios.