THE LEAST PRIOR DEVIATION QUASI-NEWTON UPDATE

We propose a new choice for the parameter in the Broyden class and der ive and discuss properties of the resulting self-complementary quasi-N ewton update. Our derivation uses a variational principle that minimiz es the extent to which the quasi-Newton relation is violated on a prio r step. We discuss the merits of the variational principle used here v is-a-vis the other principle in common use, which minimizes deviation from the current Hessian or Hessian inverse approximation in an approp riate Frobenius matrix norm. One notable advantage of our principle is an inherent symmetry that results in the same update being obtained r egardless of whether the Hessian matrix or the inverse Hessian matrix is updated. We describe the relationship of our update to the BFGS, SR I and DFP updates under particular assumptions on line search accuracy , type of function being minimized (quadratic or nonquadratic) and nor m used in the variational principle.Some considerations concerning imp lementation are discussed and we also give a numerical illustration ba sed on an experimental implementation using MATLAB.