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

TL;DR

This paper introduces a Scholtes-type regularization method for solving nonlinear programming problems with cardinality constraints, demonstrating convergence to local minima and applying it to robust portfolio optimization with various risk measures.

Contribution

It develops a convergence analysis for a Scholtes-type regularization method applied to cardinality-constrained problems and evaluates its performance in portfolio optimization scenarios.

Findings

The sequence of solutions converges to an S-stationary point under convexity.

The method performs well compared to other solution approaches in numerical experiments.

Application to portfolio optimization with Value-at-Risk and Conditional Value-at-Risk.

Abstract

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.