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

0489264 - ÚTIA 2019 RIV US eng J - Journal Article
Branda, Martin - Bucher, M. - Červinka, Michal - Schwartz, A.
Convergence of a Scholtes-type regularization method for cardinality-constrained optimization problems with an application in sparse robust portfolio optimization.
Computational Optimization and Applications. Roč. 70, č. 2 (2018), s. 503-530. ISSN 0926-6003. E-ISSN 1573-2894
R&D Projects: GA ČR GA15-00735S
Institutional support: RVO:67985556
Keywords : Cardinality constraints * Regularization method * Scholtes regularization * Strong stationarity * Sparse portfolio optimization * Robust portfolio optimization
OECD category: Statistics and probability
Impact factor: 1.906, year: 2018
http://library.utia.cas.cz/separaty/2018/MTR/branda-0489264.pdf

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.
Permanent Link: http://hdl.handle.net/11104/0283708