Publication detail

Convergence of a Scholtes-type regularization method for cardinality-constrained optimization problems with an application in sparse robust portfolio optimization

Author(s): RNDr. Michal Červinka Ph.D., Branda M., Bucher M., Schwartz A.
Type: Articles in journals with impact factor
Year: 2018
Number: 0
ISSN / ISBN: 0926-6003
Published in: Computational Optimization and Applications
Publishing place:
Keywords: Cardinality constraints, Regularization method, Scholtes regularization, Strong stationarity, Sparse portfolio optimization, Robust portfolio optimization
JEL codes:
Suggested Citation: Comput Optim Appl (2018) 70:503-530
Grants: GAČR 13-01930S Robust methods for nonstandard situations, their diagnostics and implementations
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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