RANK MODIFICATIONS OF SEMIDEFINITE MATRICES ASSOCIATED WITH A SECANT UPDATE FORMULA

This paper analyzes rank modification of symmetric positive definite m atrices H of the form H - M + P, where H - M denotes a step of reducin g H to a lower-rank, symmetric and positive semidefinite matrix and (H - M) + P denotes a step of restoring H - M to a symmetric positive de finite matrix. These steps and their generalizations for rectangular m atrices are fully characterized. The well-known BFGS and DFP updates u sed in Hessian and inverse Hessian approximations provided the motivat ion for the generalizations and are special cases with H and P having rank one.