Convergence of a Scholtes-type regularization method for cardinality-constrained optimization problems with an application in sparse robust portfolio optimization
- 330 Downloads
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.
KeywordsCardinality 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.
- 9.Bucher, M., Schwartz, A.: Second order optimality conditions and improved convergence results for a Scholtes-type regularization for a continuous reformulation of cardinality constrained optimization problems. Research Report. https://arxiv.org/abs/1709.01368 (2017)Google Scholar
- 10.Burdakov, O.P., Kanzow, Ch., Schwartz, A.: On a reformulation of mathematical programs with cardinality constraints. In: Gao, D.Y., Ruan, N., Amd, W., Xing, X. (eds.), Proceedings of the 3rd World Congress of Global Optimization (Huangshan, China, 2013), Springer, New York, pp. 121–140 (2015)Google Scholar
- 27.Frangioni, A., Gentile, D.C.: Mean-variance problem with minimum buy-in constraints, data and documentation. http://www.di.unipi.it/optimize/Data/MV.html
- 28.Feng, M., Mitchell, J.E., Pang, J.-S., Shen, X., Wächter, A.: Complementarity formulation of $ \ell _0 $-norm optimization problems. Technical Report, Industrial Engineering and Management Sciences, Northwestern University, Evanston, IL, USA, September (2013)Google Scholar
- 30.Gill, P.E., Murray, W., Saunders, W., Wong, E.: SNOPT 7.5 user’s manual. CCoM Technical Report, vol. 15-3, Center for Computational Mathematics, University of California, San DiegoGoogle Scholar
- 33.Gurobi Optimization, Inc.: Gurobi optimizer reference manual. http://www.gurobi.com (2016)
- 42.Markowitz, H.M.: Portfolio selection. J. Finance 7, 77–91 (1952)Google Scholar
- 54.Shapiro, A.: Topics in Stochastic Programming. CORE Lecture Series, Université catholique de Louvain, Louvain-la-nueve (2011)Google Scholar