Line searches

Fig. 5.7 A line search is used to locate the minimum in the function in the direction of the gradient.

The gradient at the minimum point obtained from the line search will be perpendicular to the previous direction. Thus, when the line search method is used to locate the minimum along the gradient then the next direction in the steepest descents algorithm will be orthogonal to the previous direction (i.e. gk Sk-i = 0)-... 

Fig. 5.8 The minimum in a line search may be found more effectively by fitting an analytical function such as a quadratic to the initial set of three points (1, 2 and 3). A better estimate of the minimum can then be found by fitting a new function to the points 1, 2 and 4 and finding its minimum. (Figure adapted from Press W H, B P Flannery,...

This formula is exact for a quadratic function, but for real problems a line search may be desirable. This line search is performed along the vector — x. . It may not be necessary to locate the minimum in the direction of the line search very accurately, at the expense of a few more steps of the quasi-Newton algorithm. For quantum mechanics calculations the additional energy evaluations required by the line search may prove more expensive than using the more approximate approach. An effective compromise is to fit a function to the energy and gradient at the current point x/t and at the point X/ +i and determine the minimum in the fitted function. 

BRS/Morning Search is available only in Europe and retrieves information from the BRS Online Service databank. BRS/Colleague provides access to the BRS Online Service databank, but it is a menu-driven on-line service designed for use by health professionals with or without on-line search experience... 

With the advent of on-line searching in the 1970s, the Derwent file was one of the first to go on-line. It had subject retrieval capabiUty by the manual and punch code systems, tide terms, IPC, and broad subject groupings called Derwent classes, whose primary function had been to allocate patents to appropriate segments of the Derwent system. By 1981, abstracts were added to the database, after which abstracts for the entire back-file were added. 

Review and include the summary of the newspaper/industry on-line search... 

An accurate line search will require several function evaluations along each search direction. Often the minimization along the line is only carried out fairly crudely, or a... 

Let II II denote the Euclidean norm and define = gk+i gk- Table I provides a chronological list of some choices for the CG update parameter. If the objective function is a strongly convex quadratic, then in theory, with an exact line search, all seven choices for the update parameter in Table I are equivalent. For a nonquadratic objective functional J (the ordinary situation in optimal control calculations), each choice for the update parameter leads to a different performance. A detailed discussion of the various CG methods is beyond the scope of this chapter. The reader is referred to Ref. [194] for a survey of CG methods. Here we only mention briefly that despite the strong convergence theory that has been developed for the Fletcher-Reeves, [195],... 

Note that with an exact line search, Hence, the... 

Essentially we need to perform a simple line search along the direction of Ak(j+I). The simplest way to do this is by approximating the objective function by a quadratic along this direction. Namely,... 

CONOPT (Drud, 1994) Reduced gradient Line search Exact and quasi-Newton... 

MINOS Reduced gradient, augmented lagrangian Line search Quasi-Newton... 

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