Search

The BFGS method is one of the most, famous quasi-Newton algorithms for unconstrained optimization. In 1984, Powell presented an example of a function of two variables that shows that the Polak Ribi[egrave;]re Polyak (PRP) conjugate gradient method and the BFGS quasi-Newt, on method may cycle around eight nonstationary points if each line search picks a local minimum that provides a reduction in the objective function. In this paper, a new technique of choosing parameters is introduced, and an example with only six cyclic points is provided. It is also noted through the examples that the BFGS method with Wolfe line searches need not converge for nonconvex objective functions. [ABSTRACT FROM AUTHOR]

We consider the rooted orienteering problem: Given a set P of n points in the plane, a starting point r ∊ P, and a length constraint B, one needs to find a path starting from r that visits as many points of P as possible and of length not exceeding B. We present a (1 - ϵ)-approximation algorithm for this problem that runs in nO(1/ϵ) time; the computed path visits at least (1 - ϵ)kopt points of P, where kopt is the number of points visited by an optimal solution. This is the first polynomial time approximation scheme for this problem. The algorithm also works in higher dimensions. [ABSTRACT FROM AUTHOR]