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

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