A new line search strategy for finding separating hyperplane in projection-based methods

A unified derivative-free projection method model for large-scale nonlinear equations with convex constraints

“…where γ k = F (x k +α k d k ) 1+ F (x k +α k d k ) . Preliminary numerical results and related comparisons in [2] show that the new scheme is very effective and can improve the efficiency of projection-based methods. Subsequently, Ou and Li [16] also gave an adaptive scheme, i.e., it computes a stepsize α k = max{τ…”

“…see [13,1,12,7,17,25,27,11,3,28] for instance. Obviously, If F is the gradient of a real-valued function f defined in R n , the condition (2) implies that d k is a sufficiently descent direction of f at x k , which plays an important role in analyzing the convergence property of gradient-related methods.…”

“…As for the choice of α k , two well known line search procedures have been proposed in projection-based methods for solving problem (1) [2]. More precisely, given a current iterate x k and a search direction d k at the k-th iteration, α k is determined, respectively as follows:…”

“…Numerical experiments indicate that the above-mentioned line search schemes I and II have different performance during solving problem (1), see [2] for details. In order to maintain their advantages and overcome their disadvantages, a new adaptive line search scheme is introduced in [2], i.e., it chooses a stepsize α k = max{τ ρ i :…”

“…It should be noted that almost all of the above-mentioned derivative-free projection methods for problem (1) require that the mapping F is monotone and Lipschitz continuous, see [13,1,12,7,17,25,27,11,3,28,2,16] for instance. However, the requirement of monotonicity or Lipschitiz continuity seems too stringent for the purpose of establishing global convergence property.…”