The NLPU solver implements an inexact line-search algorithm. Given a search direction (for example, produced by a quasi-Newton method), each iteration of the line search selects an appropriate step length , which would in some sense approximate , an optimal solution to the following problem:

Define as

During subsequent line-search iterations, objective function values , , and their gradients , are used to construct a cubic polynomial interpolation, whose minimizer over gives the next iteration step length.

An early (economical) line-search termination criterion is given by strong Wolfe conditions

where is a sufficient decrease condition constant, known as Armijo’s constant, and is a strong curvature condition constant, known as Wolfe’s constant. If is bounded below and is a descent direction at (such that ), then there is always a step length that satisfies strong Wolfe conditions.