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

arXiv: https://arxiv.org/abs/1703.10637 · HTML: https://mlsanity.com/paper/1703.10637/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Mar 2017.

## Built on by (most cited first, 1 of 1)

- [Probabilistic Iterative Hard Thresholding for Sparse Learning](https://mlsanity.com/paper/2409.01413/) (2024)

## Abstract (arXiv)

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.