A "dark cloud" hangs over numerical optimization theory for decades, namely, whether an optimization algorithm $O(\log(n))$ iteration complexity exists. "Yes", this paper answers, with a new optimization algorithm and strict theory proof. It starts with box-constrained quadratic programming (Box-QP), and many practical optimization problems fall into Box-QP. General smooth quadratic programming (QP), nonsmooth Lasso, and support vector machine (or regression) can be reformulated as Box-QP via duality theory. It is the first time to present an $O(\log(n))$ iteration complexity QP algorithm, in particular, which behaves like a "direct" method: the required number of iterations is deterministic with exact value $\left\lceil\log\left(\frac{3.125n}{\epsilon}\right)/\log(1.5625)\right\rceil$. This significant breakthrough enables us to transition from the $O(\sqrt{n})$ to the $O(\log(n))$ optimization algorithm, whose amazing scalability is particularly relevant in today's era of big data and artificial intelligence.

翻译：数十年来，数值优化理论中一直笼罩着一朵“乌云”，即是否存在一种具有 $O(\log(n))$ 迭代复杂度的优化算法。“是的”，本文回答了这一问题，并提出了一种新的优化算法及其严格的理论证明。该算法以带箱约束的二次规划（Box-QP）为起点，而许多实际优化问题均可归为Box-QP类型。通过对偶理论，一般的平滑二次规划（QP）、非光滑Lasso以及支持向量机（或回归）都可以转化为Box-QP形式。本文首次提出了一种具有 $O(\log(n))$ 迭代复杂度的QP算法，特别地，该算法表现如同“直接”方法：所需迭代次数是确定性的，其精确值为 $\left\lceil\log\left(\frac{3.125n}{\epsilon}\right)/\log(1.5625)\right\rceil$。这一重大突破使我们能够从 $O(\sqrt{n})$ 优化算法过渡到 $O(\log(n))$ 优化算法，其惊人的可扩展性在当今大数据与人工智能时代尤为重要。