A primal–dual interior point method for a novel type-2 second order cone optimization

Abstract

In this paper, we define a new, special second order cone as a typek second order cone. We focus on the case of k = 2, which can be viewed as SOCO with an additional complicating variable. For this new problem, we develop the necessary prerequisites, based on previous work for traditional SOCO. We then develop a primal-dual interior point algorithm for solving a type-2 second order conic optimization (SOCO) problem, based on a family of kernel functions suitable for this type-2 SOCO. We finally derive the following iteration bound for our framework:$$\frac{1}{\theta \kappa \gamma} \left[2N \psi\left( \frac{\varrho \left(\tau /4N\right)}{\sqrt{1-\theta}}\right)\right]^\gamma\log \frac{3N}{\epsilon}.$$