Algorithm

Applied mathematics

Computer security

Nonlinear conjugate gradient method

Convergence (economics)

Gradient descent

RADIUS

Economics

Computer science

Mathematical optimization

Counterexample

Line search

Mathematics

Path (computing)

Artificial intelligence

Geometry

Combinatorics

Point (geometry)

Programming language

Economic growth

Artificial neural network

Conjugate gradient method

Line (geometry)

Journal of Computational and Applied Mathematics

Powell (1984) and Dai (2003) constructed respectively a counterexample to show that the Polak–Ribière–Polyak (PRP) conjugate gradient algorithm fails to globally converge for nonconvex functions even when the exact line search technique is used, which implies similar failure of the weak Wolfe–Powell (WWP) inexact line search technique. Does another inexact line search technique exist that can ensure global convergence for nonconvex functions? This paper gives a positive answer to this question and proposes a modified WWP (MWWP) inexact line search technique. An algorithm is presented that uses the MWWP inexact line search technique; the next point xk+1 generated by the PRP formula is accepted if a positive condition holds, and otherwise, xk+1 is defined by a technique for projection onto a parabolic surface. The proposed PRP conjugate gradient algorithm is shown to possess global convergence under inexact line search for nonconvex functions. Numerical performance of the proposed algorithm is shown to be competitive with those of similar algorithms.

The global convergence of the Polak–Ribière–Polyak conjugate gradient algorithm under inexact line search for nonconvex functions