New zero-finders for trust-region computations
Trust-region methods are among the most popular schemes when solving optimization problems. An important part of each trust-region method is determination of a direction vector. It is a unique minimum of a quadratic function that approximates the function to be minimized on the set called the trust-region. This direction vector depends on a parameter that is determined by applying the Newton method for the computation of the zero of a certain nonlinear function representing the distance of the direction vector from the trust-region. Several known zero-finders that exhibit quadratic convergence are described in the paper. Then, the authors bring two new zero-finders that yield cubic convergence and only require insignificantly additional computational work per step than the quadratically convergent zero-finders described. It is based on the observation that the second derivative of the corresponding function can be evaluated inexpensively when the first derivative is known. The new zero-finders are first discussed in the context of determining a zero of a fairly general function with two continuous derivatives and at least one zero. Then, the application to the trust-region computation is discussed.
scientific article; zbMATH DE number 6808520
scientific article