In this paper, an interior point method (IPM) based on a new kernel function for solving linearly constrained convex optimization problems is presented. So, firstly a survey on several trigonometric kernel functions defined in literature is done and some properties of them are studied. Then some common characteristics of these functions which help us to define a new trigonometric kernel function are obtained. We generalize the growth term of the kernel function by applying a positive parameter p and rewritten the trigonometric kernel functions defined in the literature. By the help of some simple analysis tools, we show that the IPM based on the new kernel function obtains \(O\left( \sqrt{n}\log n\log \frac{n}{\epsilon }\right) \) iteration complexity bound for large-update methods. Finally, we illustrate some numerical results of performing IPMs based on the kernel functions for solving non-negative matrix factorization problems.

中文翻译：

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基于核函数的线性约束凸优化内点算法及其在非负矩阵分解中的应用

本文提出了一种基于核函数的内点法（IPM），用于求解线性约束凸优化问题。因此，首先对文献中定义的几种三角核函数进行了调查，并研究了它们的一些特性。然后获得这些函数的一些共同特征，这些特征有助于我们定义新的三角核函数。我们通过应用正参数p并重写文献中定义的三角核函数来概括核函数的增长项。通过一些简单的分析工具，我们证明了基于新内核功能的IPM获得\（O \ left（\ sqrt {n} \ log n \ log \ frac {n} {\ epsilon} \ right）\ ）迭代复杂度受大型更新方法的约束。最后，我们说明了基于内核函数执行IPM解决非负矩阵分解问题的一些数值结果。