An interior-point algorithm for linearly constrained convex optimization based on kernel function and application in non-negative matrix factorization

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.