Least Frobenius norm updating of quadratic models that satisfy interpolation conditions

Abstract.

Quadratic models of objective functions are highly useful in many optimization algorithms. They are updated regularly to include new information about the objective function, such as the difference between two gradient vectors. We consider the case, however, when each model interpolates some function values, so an update is required when a new function value replaces an old one. We let the number of interpolation conditions, m say, be such that there is freedom in each new quadratic model that is taken up by minimizing the Frobenius norm of the second derivative matrix of the change to the model. This variational problem is expressed as the solution of an (m+n+1)×(m+n+1) system of linear equations, where n is the number of variables of the objective function. Further, the inverse of the matrix of the system provides the coefficients of quadratic Lagrange functions of the current interpolation problem. A method is presented for updating all these coefficients in ({m+n}²) operations, which allows the model to be updated too. An extension to the method is also described that suppresses the constant terms of the Lagrange functions. These techniques have a useful stability property that is investigated in some numerical experiments.