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Journal Article

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

Branda Martin, Bucher M., Červinka Michal, Schwartz A.

: Computational Optimization and Applications vol.70, 2 (2018), p. 503-530

: GA15-00735S, GA ČR

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

: 10.1007/s10589-018-9985-2

: http://library.utia.cas.cz/separaty/2018/MTR/branda-0489264.pdf

(eng): 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\nproblem 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.

: BB

: 10103