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A symmetric grouped and ordered multi-secant Quasi-Newton update formula

For Quasi-Newton methods, one of the most important challenges is to find an estimate of the Jacobian matrix as close as possible to the real matrix. While in root-finding problems multi-secant methods are regularly used, in optimization, it is the symmetric methods (in particular BFGS) that are popular. Combining multi-secant and symmetric methods in one single update formula would combine their benefits. However, it can be proved that the symmetry and multi-secant property are generally not compatible. In this paper, we try to work around this impossibility and approach the combination of both properties into a single update formula. The novelty of our method is to group secant equations based on their relative importance and to order those groups. This leads to a generic formulation of a symmetric Quasi-Newton method that is as close as possible to satisfying multiple secant equations. Our new update formula is modular and can be used in different applications where multiple secant equations, coming from different sources, are available. The formulation encompasses also different existing Quasi-Newton symmetric update formulas that try to approach the multi-secant property.