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.