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 ∈ C \ {0}. We choose a point ¯s ∈ intC∗ that serves as a normalizer and consider computational properties of the normalized system F¯s : Ax = 0, ¯sT x = 1, x ∈ C. We show that the computational complexity of solving F via an interior-point method depends only on the complexity value ϑ of the barrier for C and on the symmetry of the origin in the image set H¯s := {Ax : ¯sT x = 1, x ∈ C}, where the symmetry of 0 in H¯s is sym(0,H¯s) := max{α : y ∈ H¯s -->−αy ∈ H¯s} .We show that a solution of F can be computed in O(sqrtϑ ln(ϑ/sym(0,H¯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¯s and the image set H¯s and prove the existence of a normalizer ¯s such that sym(0,H¯s) ≥ 1/m provided that F has an interior solution. We develop a methodology for constructing a normalizer ¯s such that sym(0,H¯s) ≥ 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-polynomialtime, the normalizer will yield a conic system that is solvable in O(sqrtϑ ln(mϑ)) iterations, which is strongly-polynomialtime. 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 1000 × 5000.