Yau's Affine-Normal Descent for Large-Scale Unrestricted Higher-Moment Portfolio Optimization

2026-04-28

AI Summary

This study addresses the formidable computational challenges in large-scale higher-moment portfolio optimization—specifically, the dense nonconvex quartic objective arising from incorporating mean, variance, skewness, and kurtosis, alongside high-dimensional co-skewness and co-kurtosis tensors. The authors introduce affine differential geometry to this domain for the first time, proposing a structure-aware Yau affine normal descent algorithm. This method constructs an exact sample oracle directly from the return matrix, thereby circumventing explicit formation of higher-order tensors. It integrates preconditioned conjugate gradients, a stagnation-recovery mechanism, and a quartic-structure-driven exact line search, while decoupling data geometry from preference parameters to establish regularity and convexity theory over a simplified simplex. Empirical results on 5-minute panel data of 5,440 A-shares demonstrate that, under moderate return targets, the higher-moment strategy significantly outperforms the classical mean-variance benchmark.

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.

Problem

• portfolio optimization
• higher-moment
• nonconvex optimization
• large-scale
• cokurtosis

Innovation

• affine-normal descent
• higher-moment optimization
• nonconvex quartic optimization
• tensor-free algorithm
• large-scale portfolio optimization