Abstract

We present a polynomial-time primal-dual interior-point algorithm (IPA) for solving convex quadratic optimization (CQO) problems, based on a bi-parameterized bi-hyperbolic kernel function (KF). The growth term is a combination of the classical quadratic term and a hyperbolic one depending on a parameter \(p\in[0,1],\) while the barrier term is hyperbolic and depends on a parameter \(q\geq\frac{1}{2}\sinh 2.\) Using some simple analysis tools, we prove with a special choice of the parameter \(q,\) that the worst-case iteration bound for the new corresponding algorithm is \(\textbf{O}\big{(}\sqrt{n}\log n\log\frac{n}{\epsilon}\big{)}\) iterations for large-update methods. This improves the result obtained in (Optimization 70 (8), 1703–1724 (2021)) for CQO problems and matches the currently best-known iteration bound for large-update primal-dual interior-point methods (IPMs). Numerical tests show that the parameter \(p\) influences also the computational behavior of the algorithm although the theoretical iteration bound does not depends on this parameter. To our knowledge, this is the first bi-parameterized bi-hyperbolic KF-based IPM introduced for CQO problems, and the first KF that incorporates a hyperbolic function in its growth term while all KFs existing in the literature have a polynomial growth term exepct the KFs proposed in (Optimization 67 (10), 1605–1630 (2018)) and (J. Optim. Theory Appl. 178, 935–949 (2018)) which have a trigonometric growth term.

中文翻译：

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基于新双参数化双曲核函数的凸二次优化大更新原对偶内点算法

 抽象的

我们提出了一种基于双参数化双双曲核函数 (KF) 的多项式时间原对偶内点算法 (IPA)，用于求解凸二次优化 (CQO) 问题。增长项是经典二次项和双曲项的组合，具体取决于参数 \(p\in[0,1],\)，而势垒项是双曲项，取决于参数 \(q\geq\frac {1}{2}\sinh 2.\) 使用一些简单的分析工具，我们通过参数 \(q,\) 的特殊选择证明新的相应算法的最坏情况迭代界限为 \(\textbf大更新方法的 {O}\big{(}\sqrt{n}\log n\log\frac{n}{\epsilon}\big{)}\) 次迭代。这改进了（Optimization 70 (8), 1703–1724 (2021)）中针对 CQO 问题获得的结果，并与大更新原始对偶内点方法 (IPM) 的当前最著名的迭代界限相匹配。数值测试表明，参数 \(p\) 也会影响算法的计算行为，尽管理论迭代界限不依赖于该参数。据我们所知，这是针对 CQO 问题引入的第一个基于双参数化双曲 KF 的 IPM，也是第一个在其增长项中包含双曲函数的 KF，而文献中现有的所有 KF 都具有多项式增长项，除了(Optimization 67 (10), 1605–1630 (2018)) 和 (J. Optim. Theory Appl. 178, 935–949 (2018)) 中提出的 KF 具有三角增长项。