The least prior deviation quasi-Newton update

The authors propose a new choice for the parameter in the Broyden class and derive and discuss properties of the resulting self-complementary quasi-Newton update. The derivation uses a variational principle that minimizes the extent to which the quasi-Newton relation is violated on a prior step. The authors discuss the merits of the variational principle used here vis-a-vis the other principle in common use, which minimizes deviation from the current Hessian or Hessian inverse approximation in an appropriate Frobenius matrix norm. One notable advantage of the principle is an inherent symmetry that results in the same update being obtained regardless of whether the Hessian matrix or the inverse Hessian matrix is updated. The authors describe the relationship of the update to the BFGS, SR1 and DFP updates under particular assumptions on line search accuracy, type of function being minimized (quadratic or nonquadratic) and norm used in the variational principle. Some considerations concerning implementation are discussed and the authors also give a numerical illustration based on an experimental implementation using MATLAB.