Computational Optimization and Applications

, Volume 70, Issue 2, pp 503–530 | Cite as

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

  • Martin BrandaEmail author
  • Max Bucher
  • Michal Červinka
  • Alexandra Schwartz


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.


Cardinality constraints Regularization method Scholtes regularization Strong stationarity Sparse portfolio optimization Robust portfolio optimization 



Martin Branda and Michal Červinka would like to acknowledge the support of the Czech Science Foundation (GA ČR) under the Grants GA13-01930S, GA15-00735S and GA18-05631S. The work of Alexandra Schwartz and Max Bucher is supported by the ‘Excellence Initiative’ of the German Federal and State Governments and the Graduate School of Computational Engineering at Technische Universität Darmstadt. The authors would like to express their gratitude to the anonymous referees for their valuable comments and suggestions, which led to many improvements in the manuscript.


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Authors and Affiliations

  1. 1.The Czech Academy of SciencesInstitute of Information Theory and AutomationPrague 8Czech Republic
  2. 2.Department of Probability and Mathematical Statistics, Faculty of Mathematics and PhysicsCharles UniversityPrague 8Czech Republic
  3. 3.Graduate School Computational EngineeringTechnische Universität DarmstadtDarmstadtGermany
  4. 4.Faculty of Social SciencesCharles UniversityPrague 1Czech Republic

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