The LP-Newton Method and Conic Optimization

TL;DR

This paper introduces an adaptation of the LP-Newton method for solving conic linear programs over conic boxes, extending its applicability to SOCP and SDP problems with promising efficiency, supported by experimental results.

Contribution

It generalizes the LP-Newton method to conic LPs over conic boxes and demonstrates its adaptation to SOCP and SDP, with practical implementation insights.

Findings

Low number of Newton steps needed for convergence

Effective adaptation to SOCP and SDP problems

Experimental validation shows promising results

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.