On a one-dimensional optimization problem derived from the efficiency analysis of Newton-PCG-like algorithms (Q697541)

Recently, a new type of Newton-preconditioned conjugate gradient (PCG-like algorithm is derived for solving the unconstrained optimization problem \(\min f(x)\), \(x\in\mathbb{R}^n\). The basic idea to construct the algorithms model for such kind algorithms is that the Newton equation are solved exactly by Cholesky factorization (CF) or approximately by preconditioned conjugate gradient method. Each circle of the model consists of one CF step and \(p\) PCG steps, where \(p\) is a parameter which is chosen in such a way that the efficiency of the algorithms to be maximized. It is proved that the efficiency of such kind methods is superior to that of Newton's method for the special cases where Newton's method converges with precise order 2 or \(\alpha\), respectively. This paper studies the more general case where the convergence speed of Newton's method is unknown and even Newton's method may have no fixed order, i.e. the convergence rate of Newton's method is allowed to be in some interval. In a cirle of the new Newton -- PCG -- like algorithm, at first, the approximate progress speed of CF step is calculated and thereafter the corresponding one-dimensional problem is solved to obtain the value of the parameter \(p\) to finish the following PCG step of this cirle. To overcome the difficulty associated with the necessity to solve a series of the one-dimensional optimization problems an analytic expression of the solution to the one-dimensional optimization problem with the parameter \(\alpha\) is derived.