Gradient method step length

The A for the minimization modes is determined as for the RFO method, eq. (14.8). The equation for Ays is quadratic, and by choosing the solution which is larger than 8ts it is guaranteed that the step component in this direction is along the gradient, i.e. a maximization. As for the RFO step, there is no guarantee that the total step length will be within the trust radius. 

Although there are new non-monotone methods, which do not rely strictly on the monotonicity (1), see [3].) If the descent f(xk)—f xk+i) is proportional to the step length and the norm of the current gradient,... 

The Muller-Brown method (MB) (Figure 3b) is an implicit Euler method. A step of length a is taken from 3c, and the energy is minimized under the constraint of constant step length, 3c,+i - 3c, = a. At the minimum, the residual gradient is parallel to the step. Thus the implicit Euler step can also be written as 3c, +, = 3c, - agi+1 / g,+11 = x/ +, ... 

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