Advanced methods in portfolio selection: index tracking, dominance maximization, and weights-based strategy identification

This work addresses key challenges in both academia and the financial industry, contributing to the literature by providing advanced methods and models for real-world applications in portfolio management. The first part of the research addresses the index tracking problem through the Single-Index Principal Component Analysis method. This approach identifies a small subset of assets to replicate benchmark performance, significantly reducing turnover and computational time compared to traditional optimization-based techniques. The second part of this thesis investigates sparse portfolio selection, proposing a novel risk-gain-sparsity model based on $\ell_1$-norm regularization. This framework yields an efficient portfolio that dominates the reference benchmark while maintaining a limited number of assets. The problem is solved via a projected gradient algorithm, with empirical results showing promising out-of-sample performance. Finally, we explore the capital distribution of optimal portfolios using rank-size analysis. By successfully reverse-engineering allocation strategies from their weight structures via a Random Forest classifier, we demonstrate that the geometry of capital distribution serves as a powerful tool for identifying the underlying optimization framework. All models are tested with extensive empirical analyses based on real-world data.