A Large-update Primal-dual Interior-point Algorithm for Convex Quadratic Optimization Based on a New Bi-parameterized Bi-hyperbolic Kernel Function

IF 0.8 Q2 MATHEMATICS

Lobachevskii Journal of Mathematics Pub Date : 2024-07-19 DOI:10.1134/s1995080224600560

Youssra Bouhenache, Wided Chikouche, Imene Touil

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引用次数: 0

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.

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基于新的双参数化双双曲核函数的凸二次方优化大更新原点-双内部点算法

摘要我们提出了一种基于双参数化双双曲核函数（KF）的多项式时间原始双内点算法（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，取决于一个参数（p\in[0,1]\），而障碍项是双曲项，取决于一个参数（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，对于大更新方法，新的相应算法的最坏情况迭代约束是 \(textbf{O}\big{(}\sqrt{n}\log nlog\frac{n}{\epsilon}\big{)}\) 次迭代。这改进了《优化》（Optimization 70 (8), 1703-1724 (2021)）中针对 CQO 问题得到的结果，并与目前已知的大更新原始双内点法（IPMs）的迭代约束相匹配。数值测试表明，虽然理论迭代约束并不取决于参数 \(p\)，但参数 \(p\)也会影响算法的计算行为。据我们所知，这是第一个针对 CQO 问题提出的基于双参数化双双曲 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 都有一个多项式增长项，除了在（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。

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来源期刊

Lobachevskii Journal of Mathematics MATHEMATICS-

CiteScore

1.50

自引率

42.90%

发文量

127

期刊介绍： Lobachevskii Journal of Mathematics is an international peer reviewed journal published in collaboration with the Russian Academy of Sciences and Kazan Federal University. The journal covers mathematical topics associated with the name of famous Russian mathematician Nikolai Lobachevsky (Lobachevskii). The journal publishes research articles on geometry and topology, algebra, complex analysis, functional analysis, differential equations and mathematical physics, probability theory and stochastic processes, computational mathematics, mathematical modeling, numerical methods and program complexes, computer science, optimal control, and theory of algorithms as well as applied mathematics. The journal welcomes manuscripts from all countries in the English language.