In large-scale, data-driven applications, parameters are often only known approximately due to noise and limited data samples. In this paper, we focus on high-dimensional optimization problems with linear constraints under uncertain conditions. To find high quality solutions for which the violation of the true constraints is limited, we develop a linear shrinkage method that blends random matrix theory and robust optimization principles. It aims to minimize the Frobenius distance between the estimated and the true parameter matrix, especially when dealing with a large and comparable number of constraints and variables. This data-driven method excels in simulations, showing superior noise resilience and more stable performance in both obtaining high quality solutions and adhering to the true constraints compared to traditional robust optimization. Our findings highlight the effectiveness of our method in improving the robustness and reliability of optimization in high-dimensional, data-driven scenarios.

翻译：在大规模数据驱动应用中，由于噪声和有限的数据样本，参数通常仅能近似已知。本文聚焦于不确定条件下带有线性约束的高维优化问题。为寻求高质量解并限制真实约束的违反程度，我们提出了一种融合随机矩阵理论与鲁棒优化原理的线性收缩方法。该方法旨在最小化估计参数矩阵与真实参数矩阵之间的Frobenius距离，尤其在处理数量相当且规模庞大的约束与变量时效果显著。模拟实验表明，与传统的鲁棒优化相比，这种数据驱动方法在获取高质量解和满足真实约束两方面均展现出更强的噪声鲁棒性和更稳定的表现。我们的研究突显了该方法在提升高维数据驱动场景下优化鲁棒性与可靠性的有效性。