Sparse mean–variance customer Markowitz portfolio optimization for Markov chains: a Tikhonov’s regularization penalty approach

This paper considers the subject of penalty regularized expected utilities and investigates the applicability of the method for computing the mean–variance Markowitz customer portfolio optimization problem. We penalize the large values by introducing a penalty term expressed as least-squares in order to avoid an explosive number of solutions. This penalty term is known as the Tikhonov regularization parameter. Tikhonov’s regularization is one of the most popular approaches to solve discrete ill-posed problems and, in our case, it plays a fundamental role in order to ensure the convergence to a unique portfolio solution. In this sense, we first provide the parameter conditions under which the penalty regularized expected utility of a given optimal portfolio admits a unique solution. A crucial problem concerning Tikhonov’s regularization is the proper choice of the regularization parameter because it can modify (sometimes significantly) the shape of the original functional. The main objective of this paper is to derive a method for regularization in an optimal way. For solving the problem, the parameters of the regularized poly-linear optimization problem are balanced simultaneously. Then, we prove that the original Markowitz portfolio optimization problem converges to an exact solution (with the minimal weighted norm). We consider a projection gradient method for finding the extremal points including the proof of convergence of the method. We show how to select the parameters of the algorithm in order to guarantee the convergence of the suggested procedure. Finally, we present a numerical example to illustrate the practical implications of the theoretical issues of a penalty regularized portfolio optimization problem.