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. |
