AI 中文总结
本文通过结合拉格朗日对偶性与通用链，提出随机投影的新方法，扩展至线性不等式约束，并保持对偶可行性，从而近似保持对偶界。
AI 中文摘要
随机投影是一种用于优化的预处理技术，它通过将线性等式约束替换为随机线性组合来减少约束数量，同时近似保持最优值。本文强调随机过程上确界（尤其是通用链）在解释这一现象中的作用。通过将拉格朗日对偶性与通用链相结合，我们获得了更短的证明和更精确的保证，这些保证适用于超越基于Johnson-Lindenstrauss引理的典型多面体设置。更重要的是，这一视角使我们能够将随机投影扩展到具有线性不等式约束的问题。尽管随机投影的不等式约束通常既不是原始可行域的松弛也不是限制，但我们表明随机投影仍然可以近似保持对偶界。关键在于一种新的随机投影对偶证书的构造，该构造保持对偶可行性，并不同于现有受Johnson-Lindenstrauss引理启发的构造。
英文摘要
Random projection is a preprocessing technique for optimization that reduces the number of linear equality constraints by replacing them with random linear combinations while approximately preserving the optimal value. In this paper, we emphasize the role of suprema of random processes, and in particular generic chaining, in explaining this phenomenon. By combining Lagrangian duality with generic chaining, we obtain shorter proofs and sharper guarantees that apply beyond the typical polyhedral settings based on the Johnson--Lindenstrauss lemma. More importantly, this perspective allows us to extend random projection to problems with linear inequality constraints. Although randomly projected inequality constraints generally define neither a relaxation nor a restriction of the original feasible region, we show that random projection can nevertheless approximately preserve the dual bound. The key is a new construction for randomly projecting dual certificates that preserves dual feasibility and differs from existing constructions motivated by the Johnson--Lindenstrauss lemma.
