Line search methods with guaranteed asymptotical convergence to an improving local optimum of multimodal functions

This paper considers line search optimization methods using a mathematical framework based on the simple concept of a v-pattern and its properties. This framework provides theoretical guarantees on preserving, in the localizing interval, a local optimum no worse than the starting point. Notably, the framework can be applied to arbitrary unidimensional functions, including multimodal and infinitely valued ones. Enhanced versions of the golden section, bisection and Brent’s methods are proposed and analyzed within this framework: they inherit the improving local optimality guarantee. Under mild assumptions the enhanced algorithms are proved to converge to a point in the solution set in a finite number of steps or that all their accumulation points belong to the solution set.