The LP-Newton Method and Conic Optimization

Francesco Silvestri, Gerhard Reinelt

Published 2016-11-28Version 1

We propose that the LP-Newton method can be used to solve conic LPs over a 'conic interval', whenever linear optimization over an unconstrained conic interval is easy. In particular, if $\leq_\mathcal{K}$ is the partial order induced by a proper convex cone $\mathcal{K}$, then conic LPs over $[l,u]_{\mathcal{K}}=\{ l\leq_\mathcal{K} x\leq_\mathcal{K} u\}$ can be solved whenever optimizing a linear function over $[l,u]_{\mathcal{K}}$ is fast. This generalizes the result for the case of $\mathcal{K}=\mathbb{R}^n_+$ that was originally proposed for using the method. Specifically, we show how to adapt this method for both SOCP and SDP problems and illustrate the method with a few experiments.