Convergence Properties of a Class of Rank-two Updates
Paul T. Boggs, Jon W. Tolle · SIAM Journal on Optimization · 1994
Many optimization algorithms generate, at each iteration, a pair $( x_k ,H_k )$ consisting of an approximation to the solution $x_k $ and a Hessian matrix approximation $H_k $ that contains local second-order information about the problem. Much is known about the convergence of $x_k $ to the solution of the problem, but relatively little is known about the behavior of the sequence of matrix approximations. The sequence $\{ H_k \}$, generated by the extended Broyden class of updating schemes independently of the optimization setting in which they are used, is analyzed. Various conditions under which convergence is assured are derived, and the structure of the limits is delineated. Rates of convergence are also obtained. These results extend and clarify those already in the literature.