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**Branda, M. and Bucher, Max and Červinka, M. and Schwartz, Alexandra** (2018):

*Convergence of a Scholtes-type Regularization Method for Cardinality-Constrained Optimization Problems with an Application in Sparse Robust Portfolio Optimization.*

In: Computational Optimization and Applications, 70 (2), pp. 503-530. ISSN 1573-2894,

DOI: 10.1007/s10589-018-9985-2,

[Article]

## Abstract

We consider general nonlinear programming problems with cardinality constraints. By relaxing the binary variables which appear in the natural mixed-integer programming formulation, we obtain an almost equivalent nonlinear programming problem, which is thus still difficult to solve. Therefore, we apply a Scholtes-type regularization method to obtain a sequence of easier to solve problems and investigate the convergence of the obtained KKT points. We show that such a sequence converges to an S-stationary point, which corresponds to a local minimizer of the original problem under the assumption of convexity. Additionally, we consider portfolio optimization problems where we minimize a risk measure under a cardinality constraint on the portfolio. Various risk measures are considered, in particular Value-at-Risk and Conditional Value-at-Risk under normal distribution of returns and their robust counterparts under moment conditions. For these investment problems formulated as nonlinear programming problems with cardinality constraints we perform a numerical study on a large number of simulated instances taken from the literature and illuminate the computational performance of the Scholtes-type regularization method in comparison to other considered solution approaches: a mixed-integer solver, a direct continuous reformulation solver and the Kanzow--Schwartz regularization method, which has already been applied to Markowitz portfolio problems.

Item Type: | Article |
---|---|

Erschienen: | 2018 |

Creators: | Branda, M. and Bucher, Max and Červinka, M. and Schwartz, Alexandra |

Title: | Convergence of a Scholtes-type Regularization Method for Cardinality-Constrained Optimization Problems with an Application in Sparse Robust Portfolio Optimization |

Language: | English |

Abstract: | We consider general nonlinear programming problems with cardinality constraints. By relaxing the binary variables which appear in the natural mixed-integer programming formulation, we obtain an almost equivalent nonlinear programming problem, which is thus still difficult to solve. Therefore, we apply a Scholtes-type regularization method to obtain a sequence of easier to solve problems and investigate the convergence of the obtained KKT points. We show that such a sequence converges to an S-stationary point, which corresponds to a local minimizer of the original problem under the assumption of convexity. Additionally, we consider portfolio optimization problems where we minimize a risk measure under a cardinality constraint on the portfolio. Various risk measures are considered, in particular Value-at-Risk and Conditional Value-at-Risk under normal distribution of returns and their robust counterparts under moment conditions. For these investment problems formulated as nonlinear programming problems with cardinality constraints we perform a numerical study on a large number of simulated instances taken from the literature and illuminate the computational performance of the Scholtes-type regularization method in comparison to other considered solution approaches: a mixed-integer solver, a direct continuous reformulation solver and the Kanzow--Schwartz regularization method, which has already been applied to Markowitz portfolio problems. |

Journal or Publication Title: | Computational Optimization and Applications |

Journal volume: | 70 |

Number: | 2 |

Uncontrolled Keywords: | Mathematics - Optimization and Control (math.OC) |

Divisions: | Exzellenzinitiative Exzellenzinitiative > Graduate Schools Exzellenzinitiative > Graduate Schools > Graduate School of Computational Engineering (CE) 04 Department of Mathematics 04 Department of Mathematics > Optimization |

Date Deposited: | 11 Sep 2017 12:51 |

DOI: | 10.1007/s10589-018-9985-2 |

Official URL: | https://doi.org/10.1007/s10589-018-9985-2 |

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