Supermemory gradient methods for monotone nonlinear equations with convex constraints

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

“…It should be pointed out that for the existing derivative-free projection methods for solving the problem (1), Assumption A2 or A2' can be verified to be always satisfied, see [13,1,12,7,17,25,27,11,3,28,2,16] for example.…”

“…Since the derivative-free projection methods do not involve any matrix computation and storage, they are particularly effective for solving large-scale nonlinear system of equations. Therefore, this kind of method has attracted much more attention and many numerical methods have been proposed, see for example [13,1,12,7,17,25,27,11,3,28] and references therein.…”

“…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.…”

“…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.…”

“…As is well known, the analysis on the convergence rate is also important for numerical optimization algorithms. However, most of the existing derivative-free projection methods for problem (1) only discuss their global convergence, while the convergence rate is not analyzed, see [1,17,25,27,11,3,28,2] for example. In addition, even for those algorithms in which the convergence rate has been analyzed, the obtained result is only that the distance sequence {dist(x k , X * )} converges to zero with a Q-linear rate, rather than the sequence {x k } itself converges to a solution of problem (1) with a Q-linear rate, see [13,12,7] for example.…”