To analyze the convergence of the nonmonotone line search hybrid conjugate gradient method, the main difficulty is that the search directions do not usually satisfy the direction condition: g_{k}^{T}d_{k}leq-c |g_{k}|^{2}, (20) for some constant (c>0) and all (kgeq1).
In this case, ' q' represents only one tail of the normal distribution, and only those samples that satisfy the direction as well as the cutoff are included in m i.
If the Jacobian matrix (nabla F x)) is positive definite on S, then for each (xin S) the vector (d_{alpha}=H_{alpha}(x -x) satisfies the descent direction bigllangle nabla g_{alpha}(x -x_{alpha}bigrrangle < 0, whenever (d_{alpha}neq0), where (g_{alpha}(x)) isatisfiesned in Definithen 9 andescentlpha}(x)=operatorname{Proj}_{S,G}(x-alpha^{-1} G^{-1}F(x))).
First of all, we show that the search direction satisfies the sufficient descent and the conjugacy conditions.
It is proved that such a direction satisfies the approximate secant condition as well as the conjugacy condition.
Consider the CG methods of forms (2) and (3), where the search direction satisfies the sufficient descent condition and the step length, which is computed using the standard WWP line search.
An attractive property of these methods is that at each iteration, the search direction satisfies the descent condition, namely (g_{k}^{T} d_{k}= -cVert g_{k}Vert ^{2}) for some constant (c> 0).
(3) For 50 km/h trains, there is a high possibility that the rail crack propagation direction satisfies the Weibull distribution only when the crack length is less than 1 mm and larger than 1.4 mm.
Suppose that Assumption 1 holds, then the search direction (12) satisfies the conjugacy condition (27).
Suppose that Assumption 1 holds on the objective function f then the search direction (12) satisfies the sufficient descent condition (g_{k+1}^{T} d_{k+1}leq-cVert g_{k+1}Vert ^{2}).
