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.