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

In this paper we study the homogeneous conic system F : Ax = 0, x ∈ C \ {0}. We choose a point s̄ ∈ intC∗ that serves as a normalizer and consider computational properties of the normalized system Fs̄ : Ax = 0, s̄T 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 Hs̄ := {Ax : s̄T x = 1, x ∈ C}, where the symmetry of 0 in Hs̄ is sym(0, Hs̄) := max{α : y ∈ Hs̄ ⇒ −αy ∈ Hs̄} . We show that a solution of F can be computed in O( √ θ ln(θ/sym(0, Hs̄)) 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 Fs̄ and the image set Hs̄ and prove the existence of a normalizer s̄ such that sym(0, Hs̄) ≥ 1/m provided that F has an interior solution. We develop a methodology for constructing a normalizer s̄ such that sym(0, Hs̄) ≥ 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( √ θ 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.