A Newton-type univariate optimization algorithm for locating the nearest extremum

This paper introduces an algorithm for univariate optimization using linear lower bounding functions (LLBFs). An LLBF over an interval is a linear function which lies below the given function over an interval and matches the given function at one end point of the interval. We first present an algorithm using LLBFs for finding the nearest root of a function in a search direction. When the root-finding method is applied to the derivative of an objective function, it is an optimization algorithm which guarantees to locate the nearest extremum along a search direction. For univariate optimization, we show that this approach is a Newton-type method, which is globally convergent with superlinear convergence rate. The applications of this algorithm to global optimization and other optimization problems are also discussed.