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

Published

Journal Article

In this paper we study the homogeneous conic system F: Ax = 0, x in C setminus 0 . We choose a point in ∫ C that serves as a normalizer and consider computational properties of the normalized system Fs : Ax = 0, bar s-T x = 1, x ∈ C . We show that the computational complexity of solving F via an interior-point method depends only on the complexity value of the barrier for C and on the symmetry of the origin in the image set Hs := {Ax s} Tx = 1, x in C, where the symmetry of 0 in Hs is (0, Hs) := α : y Hs → -α y in H. We show that a solution of F can be computed in O(√θ In (θ/sym(0, Hs)) 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 Fs and the image set Hs and prove the existence of a normalizer s such that sym(0, Hs) ≥ 1/m provided that F has an interior solution. We develop a methodology for constructing a normalizer s such that sym (0, Hs) ≥ 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 equation pesented iterations, which is trongly-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. © 2007 Springer-Verlag.

### Full Text

### Duke Authors

### Cited Authors

- Belloni, A; Freund, RM

### Published Date

- May 1, 2009

### Published In

### Volume / Issue

- 118 / 2

### Start / End Page

- 279 - 299

### Electronic International Standard Serial Number (EISSN)

- 1436-4646

### International Standard Serial Number (ISSN)

- 0025-5610

### Digital Object Identifier (DOI)

- 10.1007/s10107-007-0192-7

### Citation Source

- Scopus