Paper: arXiv 2604.25378

Authors: Ya-Juan Wang, Yi-Shuai Niu, Artan Sheshmani, Shing-Tung Yau

Abstract

Unrestricted mean-variance-skewness-kurtosis portfolio optimization can capture asymmetry and tail risk, but sample-moment formulations become computationally impractical when the asset universe is large: they produce dense nonconvex quartic objectives with prohibitive coskewness and cokurtosis tensors and anisotropic, ill-conditioned level sets. We develop a structure-exploiting algorithm based on Yau’s affine-normal descent that follows affine-normal directions of the current level set while working directly with the return matrix. The method avoids explicit higher-order tensors and exploits the quartic structure for exact sample oracles, derivative evaluation, and exact line search. We also provide theory for the reduced simplex formulation, including regularity and convexity conditions that separate data-map geometry from investor preference coefficients. Computational results show a clear implementation split: a direct configuration is effective on the standard small benchmark, whereas a preconditioned conjugate-gradient configuration with stall recovery becomes the preferred large-scale implementation by the upper end of the hundreds and remains competitive as the asset universe moves into the thousands. On a 5-minute A-share panel with 5,440 stocks, the method makes direct full-universe comparisons with exact mean-variance portfolios feasible and shows on the baseline split that the incremental value of higher moments is strongest at moderate return targets.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 8.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper provides a rigorous theoretical derivation using Yau’s affine-normal descent on quartic objective functions while simultaneously demonstrating practical scalability through a 5,440-asset empirical test on real-world market data.

Research Flowchart

  flowchart TD
    A[<b>Research Goal:</b> Efficient High-Moment Portfolio Optimization] --> B[<b>Inputs:</b> Return matrix & Higher-order moments]
    B --> C[<b>Methodology:</b> Yau's Affine-Normal Descent]
    C --> D[<b>Computational Process:</b> Tensor-free & Quartic structure-exploiting]
    D --> E[<b>Configurations:</b> Direct vs. Preconditioned Conjugate-Gradient]
    E --> F[<b>Outcomes:</b> Scalable to 5,000+ assets & Higher-moment utility verified]