Convergence of a Scholtes-type Regularization Method for Cardinality-Constrained Optimization Problems with an Application in Sparse Robust Portfolio Optimization

摘要

We consider general nonlinear programming problems with cardinalityconstraints. By relaxing the binary variables which appear in the naturalmixed-integer programming formulation, we obtain an almost equivalent nonlinearprogramming problem, which is thus still difficult to solve. Therefore, weapply a Scholtes-type regularization method to obtain a sequence of easier tosolve problems and investigate the convergence of the obtained KKT points. Weshow that such a sequence converges to an S-stationary point, which correspondsto a local minimizer of the original problem under the assumption of convexity. Additionally, we consider portfolio optimization problems where we minimize arisk measure under a cardinality constraint on the portfolio. Various riskmeasures are considered, in particular Value-at-Risk and ConditionalValue-at-Risk under normal distribution of returns and their robustcounterparts under moment conditions. For these investment problems formulatedas nonlinear programming problems with cardinality constraints we perform anumerical study on a large number of simulated instances taken from theliterature and illuminate the computational performance of the Scholtes-typeregularization method in comparison to other considered solution approaches: amixed-integer solver, a direct continuous reformulation solver and theKanzow-Schwartz regularization method, which has already been applied toMarkowitz portfolio problems.