TU Darmstadt / ULB / TUbiblio

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

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, pp. 503-530, 70, (2), ISSN 1573-2894, DOI: 10.1007/s10589-018-9985-2, [Online-Edition: https://doi.org/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
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
Export:

Optionen (nur für Redakteure)

View Item View Item