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

Julio B. Clempner, Alexander S. Poznyak

Research output: Contribution to journalArticle

4 Citations (Scopus)

### Abstract

© 2018, Springer Science+Business Media, LLC, part of Springer Nature. 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.
Original language American English 383-417 341 Optimization and Engineering https://doi.org/10.1007/s11081-018-9374-9 Published - 1 Jun 2018

### Fingerprint

Portfolio Optimization
Markov processes
Penalty
Markov chain
Regularization
Customers
Optimization Problem
Expected Utility
Regularization Parameter
Extremal Point
Weighted Norm
Optimal Portfolio
Linear Optimization
Tikhonov Regularization
Number of Solutions
Ill-posed Problem
Term
Unique Solution

### Cite this

title = "Sparse mean–variance customer Markowitz portfolio optimization for Markov chains: a Tikhonov’s regularization penalty approach",
abstract = "{\circledC} 2018, Springer Science+Business Media, LLC, part of Springer Nature. 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.",
author = "Clempner, {Julio B.} and Poznyak, {Alexander S.}",
year = "2018",
month = "6",
day = "1",
doi = "10.1007/s11081-018-9374-9",
language = "American English",
pages = "383--417",
journal = "Optimization and Engineering",
issn = "1389-4420",
publisher = "Springer Netherlands",

}

In: Optimization and Engineering, 01.06.2018, p. 383-417.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Sparse mean–variance customer Markowitz portfolio optimization for Markov chains: a Tikhonov’s regularization penalty approach

AU - Clempner, Julio B.

AU - Poznyak, Alexander S.

PY - 2018/6/1

Y1 - 2018/6/1

N2 - © 2018, Springer Science+Business Media, LLC, part of Springer Nature. 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.

AB - © 2018, Springer Science+Business Media, LLC, part of Springer Nature. 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.

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