Solving Structured Convex Quadratic Programs by Interior Point Methods with Application to Support Vector Machines and Portfolio Optimization∗

(P ) s.t. Ax = b, 0 ≤ x ≤ u, where the symmetric positive semidefinite matrix Q ∈ Rn×n, the rank m matrix A ∈ Rm×n, c ∈ Rn, u ∈ Rn and b ∈ Rm are the data for the problem, and x ∈ Rn is the vector of variables. Three aspects of this topic are explored. The first concerns a property of the Cholesky factorization of the normal equation matrix that arises at each iteration of an IPM applied to problem (P ). In [11] it was shown that when linear programming (LP) problems are solved by an IPM, the unit lower triangular matrix L in the Cholesky factorization LΛL T of the normal equations matrix remains uniformly bounded (in norm) as the iterates converge to the solution. This result holds even if the condition number of the normal equations matrix becomes unbounded. In [12] this result was extended to the case of second-order cone (SOCP) programming under the assumption that the IPM iterates stay in a neighborhood of the central path. Here we extend the LP result (without any assumptions) to the case of convex QPs. ∗Research supported in part by NSF Grant DMS 01-04282, ONR Grant N00014-03-0514 and DOE Grant GE-FG0192ER-25126