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PROJECTIVE RENORMALIZATION FOR IMPROVING THE BEHAVIOR OF A HOMOGENEOUS CONIC LINEAR System
, 2007
"... 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, ¯s T x =1, x ∈ C. We show that the computational complexity of solving F via an interiorpo ..."
Abstract

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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, ¯s T x =1, x ∈ C. We show that the computational complexity of solving F via an interiorpoint 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: ¯s T 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 ( √ ϑ ln(ϑ/sym(0,H¯s)) interiorpoint iterations. In order to improve the theoretical and practical computation of a solution of F, we next present a general theory for projective renormalization 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 stronglypolynomialtime, the normalizer will yield a conic system that is solvable in O ( √ ϑ ln(mϑ)) iterations, which is stronglypolynomialtime. 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 renormalization 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 poorlybehaved problem instances of dimension 1000 × 5000.