Developing a symmetrical version of the quasi-newton least square algorithm

The class of Least Change Secant Update Quasi-Newton (LCSU QN) methods is often used for root finding or for optimization problems. The particularity of these methods is that the estimated Jacobian of the system is chosen satisfying a secant equation and such that it is the closest possible to the previous Jacobian. In this study, we start from a particular LCSU QN method, Quasi-Newton Least Squares (QN-LS), which has been widely used to solve interaction problems recast as a root finding problem f(x)=0. This method imposes that multiple secant equations are respected. We turn our attention to the use of QN-LS on a new problem: min g(x). If f(x) is the gradient of g(x) then the problem again becomes that of solving f(x)=0. The Jacobian of f(x), i.e. Hessian of g(x), is now symmetrical, something that is not taken in consideration in the original QN-LS method. Given that QN-LS has similarities with Broyden’s methods and given that Broyden’s methods have been developed for symmetrical Jacobians, we consider the possibilities of developing a symmetrical variant of QN-LS. We prove that this symmetrical variant only exists under limited conditions. We will therefore explore three different strategies in order to find an update formula for the estimated Jacobian, which fulfills as much as possible the symmetry and the multiple secants conditions.