Scalable Mean-Variance Portfolio Optimization via Subspace Embeddings and GPU-Friendly Nesterov-Accelerated Projected Gradient

Abstract: We develop a sketch-based factor reduction and a Nesterov-accelerated projected gradient algorithm (NPGA) with GPU acceleration, yielding a doubly accelerated solver for large-scale constrained mean-variance portfolio optimization. Starting from the sample covariance factor $L$, the method combines randomized subspace embedding, spectral truncation, and ridge stabilization to construct an effective factor $L_{eff}$. It then solves the resulting constrained problem with a structured projection computed by scalar dual search and GPU-friendly matrix-vector kernels, yielding one computational pipeline for the baseline, sketched, and Sketch-Truncate-Ridge (STR)-regularized models. We also establish approximation, conditioning, and stability guarantees for the sketching and STR models, including explicit $O(\varepsilon)$ bounds for the covariance approximation, the optimal value error, and the solution perturbation under $(\varepsilon,δ)$-subspace embeddings. Experiments on synthetic and real equity-return data show that the method preserves objective accuracy while reducing runtime substantially. On a 5440-asset real-data benchmark with 48374 training periods, NPGA-GPU solves the unreduced full model in 2.80 seconds versus 64.84 seconds for Gurobi, while the optimized compressed GPU variants remain in the low-single-digit-second regime. These results show that the full dense model is already practical on modern GPUs and that, after compression, the remaining bottleneck is projection rather than matrix-vector multiplication.