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PROJECTIVE PRECONDITIONERS FOR IMPROVING THE BEHAVIOR OF A HOMOGENEOUS CONIC LINEAR System
"... In this paper we present a general theory for transforming a normalized homogeneous conic system F: Ax = 0, ¯s T x = 1, x ∈ C to an equivalent system via projective transformation induced by the choice of a point ˆv in the set H ◦ ¯s = {v: ¯s − AT v ∈ C ∗}. Such a projective transformation serves to ..."
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In this paper we present a general theory for transforming a normalized homogeneous conic system F: Ax = 0, ¯s T x = 1, x ∈ C to an equivalent system via projective transformation induced by the choice of a point ˆv in the set H ◦ ¯s = {v: ¯s − AT v ∈ C ∗}. Such a projective transformation serves to precondition the conic system into a system that has both geometric and computational properties with certain guarantees. We characterize both the geometric behavior and the computational behavior of the transformed system as a function of the symmetry of ˆv in H ◦ ¯s as well as the complexity parameter ϑ of the barrier for C. Under the assumption that F has an interior solution, H ◦ ¯s must contain a point v whose symmetry is at least 1/m; if we can find a point whose symmetry is Ω(1/m) then we can projectively transform the conic system to one whose geometric properties and computational complexity will be stronglypolynomialtime in m and ϑ. We present a method for generating such a point ˆv based on sampling and on a geometric random walk on H ◦ ¯s with associated complexity and probabilistic analysis. 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 preconditioning 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.
On Two Measures of Problem Instance Complexity and their Correlation with the Performance of SeDuMi on SecondOrder Cone Problems ∗
, 2004
"... We evaluate the practical relevance of two measures of conic convex problem complexity as applied to secondorder cone problems solved using the homogeneous selfdual (HSD) embedding model in the software SeDuMi. The first measure we evaluate is Renegar’s databased condition measure C(d), and the s ..."
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We evaluate the practical relevance of two measures of conic convex problem complexity as applied to secondorder cone problems solved using the homogeneous selfdual (HSD) embedding model in the software SeDuMi. The first measure we evaluate is Renegar’s databased condition measure C(d), and the second measure is a combined measure of the optimal solution size and the initial infeasibility/optimality residuals denoted by S (where the solution size is measured in a norm that is naturally associated with the HSD model). We constructed a set of 144 secondorder cone test problems with widely distributed values of C(d) andS and solved these problems using SeDuMi. For each problem instance in the test set, we also computed estimates of C(d) (usingPeña’s method) and computed S directly. Our computational experience indicates that ∗This research has been partially supported through the MITSingapore Alliance.
PROJECTIVE PRECONDITIONERS FOR IMPROVING THE BEHAVIOR OF A HOMOGENEOUS CONIC LINEAR
, 2005
"... Abstract. In this paper we present a general theory for transforming a normalized homogeneous conic system F: Ax =0, ¯s T x =1, x ∈ C to an equivalent system via projective transformation induced by the choice of a point ˆv in the set H ◦ ¯s = {v:¯s − AT v ∈ C ∗}. Such a projective transformation se ..."
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Abstract. In this paper we present a general theory for transforming a normalized homogeneous conic system F: Ax =0, ¯s T x =1, x ∈ C to an equivalent system via projective transformation induced by the choice of a point ˆv in the set H ◦ ¯s = {v:¯s − AT v ∈ C ∗}. Such a projective transformation serves to precondition the conic system into a system that has both geometric and computational properties with certain guarantees. We characterize both the geometric behavior and the computational behavior of the transformed system as a function of the symmetry of ˆv in H ◦ ¯s as well as the complexity parameter ϑ of the barrier for C. Under the assumption that F has an interior solution, H ◦ ¯s must contain a point v whose symmetry is at least 1/m; if we can find a point whose symmetry is Ω(1/m) then we can projectively transform the conic system to one whose geometric properties and computational complexity will be stronglypolynomialtime in m and ϑ. We present a method for generating such a point ˆv based on sampling and on a geometric random walk on H ◦ ¯s with associated complexity and probabilistic analysis. 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 preconditioning 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. 1.