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.

翻译：无约束均值-方差-偏度-峰度投资组合优化能够捕捉尾部风险和非对称性，但当资产空间规模较大时，基于样本矩的模型在计算上变得不可行：它们会产生稠密的非凸四次目标函数，并伴随难以处理的协偏度和协峰度张量，以及各向异性、病态的水平集。我们提出了一种基于姚氏仿射-正态下降法的结构利用算法，该算法沿当前水平集的仿射-正态方向迭代，并直接对收益率矩阵进行操作。该方法避免了显式的高阶张量，并利用四次结构实现精确样本预言、导数计算及精确线搜索。我们还为简化的单纯形形式提供了理论分析，包括分离数据-映射几何与投资者偏好系数的正则性与凸性条件。计算结果表明存在清晰的实现分界线：直接配置在标准小型基准测试中表现有效，而带有停滞恢复的预条件共轭梯度配置在数百量级上端成为优选的大规模实现，并在资产空间扩展至数千量级时仍保持竞争力。在包含5440只股票的5分钟A股面板数据上，该方法使得全样本精确均值-方差投资组合的直接比较变得可行，且基线分析显示，高阶矩的增量价值在中等收益率目标处最为显著。