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Best-fit thrust network analysis
Published in Sigrid Adriaenssens, Philippe Block, Diederik Veenendaal, Chris Williams, Shell Structures for Architecture, 2014
Tom Van Mele, Daniele Panozzo, Olga Sorkine-Hornung, Philippe Block
Strong Wolfe conditions ensure that the step length reduces the objective function ‘sufficiently’, when solving an unconstrained minimization problem using an inexact line search algorithm. Strong Wolfe conditions ensure convergence of the gradient to zero.
Global convergence of three-term conjugate gradient methods on general functions under a new inexact line search strategy
Published in Engineering Optimization, 2022
Alireza Hosseini Dehmiry, Maryam Kargarfard
In CG and 3TCG algorithms, the termination conditions are frequently based on some version of the Wolfe conditions. Recall from Dai and Y. Yuan (1996) that the generalized Wolfe conditions are where and . The special cases and correspond to the strong and standard Wolfe conditions, respectively. The first condition in (7) is known as the Armijo condition, which ensures a substantial drop in the value of the objective function, while the second is known as the curvature condition, which ensures that a short step length is not acceptable.
Analysis of the gradient method with an Armijo–Wolfe line search on a class of non-smooth convex functions
Published in Optimization Methods and Software, 2020
In this section we continue to assume that f and are defined by (1) and (4), respectively, with , and that and are defined as earlier. However, unlike in the previous section, we now assume that is generated by a specific line search, namely the one given in Algorithm 1, which is taken from [16, p. 147] and is a specific realization of the line searches described implicitly in [22] and explicitly in [12]. Since the line search function is locally Lipschitz and bounded below, it follows, as shown in [16], that at any stage during the execution of Algorithm 1, the interval must always contain a set of points t with non-zero measure satisfying and , and furthermore, the line search must terminate at such a point. This defines the point . A crucial aspect of Algorithm 1 is that, in the ‘while’ loop, the Armijo condition is tested first and the Wolfe condition is then tested only if the Armijo condition holds.