The LP-Newton Method and Conic Optimization

Abstract

We propose that the LP-Newton method can be used to solve conic LPs over a conic box, whenever linear optimization over an otherwise unconstrained conic box is easy. In particular, if $\leq_\mathcal{K}$ is the partial order induced by a proper convex cone $\mathcal{K}$, then optimizing a linear function over the intersection of $[{l},{u}]_\mathcal{K}=\{{l}\leq_\mathcal{K} {x}\leq_\mathcal{K}{u}\}$ and an affine subspace can be done with this method whenever optimizing a linear function over $[{l},{u}]_{\mathcal{K}}$ is efficient. 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. While the approach is promising due to the low amount of Newton steps needed, solving the minimum-norm-point problem involved in the Newton step with a Frank-Wolfe algorithm is not advisable.