Branda, M. ; Bucher, Max ; Červinka, M. ; 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)
doi: 10.1007/s10589-018-9985-2
Artikel, Bibliographie

Kurzbeschreibung (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.

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