Projective re-normalization for improving the behavior of a homogeneous conic linear system

Abstract

In this paper we study the homogeneous conic system $F: Ax = 0, x \in C\setminus \{0\}$ . We choose a point $\bar s \in {\rm int}C^*$ that serves as a normalizer and consider computational properties of the normalized system $F_{\bar s} : Ax = 0, \bar s^T x = 1, x \in C$ . We show that the computational complexity of solving F via an interior-point method depends only on the complexity value $\vartheta$ of the barrier for C and on the symmetry of the origin in the image set $H_{\bar s} := \{Ax {\bar s}^Tx = 1, x \in C\}$ , where the symmetry of 0 in $H_{\bar s}$ is

$$ {\rm sym}(0, H_{\bar s}) := {\rm max}\{\alpha : y \in H_{\bar s} \Rightarrow -\alpha y \in H_{\bar s} \} .$$

We show that a solution of F can be computed in $O(\sqrt{\vartheta}{\rm ln}(\vartheta/{\rm sym}(0, H_{\bar s}))$ interior-point iterations. In order to improve the theoretical and practical computation of a solution of F, we next present a general theory for projective re-normalization of the feasible region $F_{\bar s}$ and the image set $H_{\bar s}$ and prove the existence of a normalizer ${\bar s}$ such that ${\rm sym}(0,H_{\bar s}) \ge 1/m$ provided that F has an interior solution. We develop a methodology for constructing a normalizer ${\bar s}$ such that ${\rm sym}(0, H_{\bar s}) \ge 1/m$ with high probability, based on sampling on a geometric random walk with associated probabilistic complexity analysis. While such a normalizer is not itself computable in strongly-polynomial-time, the normalizer will yield a conic system that is solvable in $O(\sqrt{\vartheta}{\rm ln}(m\vartheta))$ iterations, which is strongly-polynomial-time. Finally, we implement this methodology on randomly generated homogeneous linear programming feasibility problems, constructed to be poorly behaved. Our computational results indicate that the projective re-normalization methodology holds the promise to markedly reduce the overall computation time for conic feasibility problems; for instance we observe a 46% decrease in average IPM iterations for 100 randomly generated poorly-behaved problem instances of dimension 1,000  ×  5,000.