Line search strategy

The simplest procedure is merely to assume reasonable values for A and to make plots according to Eq. (2-52). That value of A yielding the best straight line is taken as the correct value. (Notice how essential it is that the reaction be accurately first-order for this method to be reliable.) Williams and Taylor have shown that the standard deviation about the line shows a sharp minimum at the correct A . Holt and Norris describe an efficient search strategy in this procedure, using as their criterion minimization of the weighted sum of squares of residuals. (Least-squares regression is treated later in this section.)... 

For process optimization problems, the sparse approach has been further developed in studies by Kumar and Lucia (1987), Lucia and Kumar (1988), and Lucia and Xu (1990). Here they formulated a large-scale approach that incorporates indefinite quasi-Newton updates and can be tailored to specific process optimization problems. In the last study they also develop a sparse quadratic programming approach based on indefinite matrix factorizations due to Bunch and Parlett (1971). Also, a trust region strategy is substituted for the line search step mentioned above. This approach was successfully applied to the optimization of several complex distillation column models with up to 200 variables. 

Global strategies for minimization are needed whenever the current estimate of the minimizer is so far from x that the local model is not a good approximation to fix) in the neighborhood of x. Three methods are considered in this section the quadratic model with line search, trust region (restricted second-order) minimization and rational function (augmented Hessian) minimization. 

A line search consists of an approximate one-dimensional minimization of the objective function along the computed direction p. This produces an acceptable step X and a new iterate xk + Xp. Function and gradient evaluations of the objective function are required in each line search iteration. In contrast, the trust region strategy minimizes approximately a local quadratic model of the function using current Hessian information. An optimal step that lies within... 

Once a new X and corresponding trial point x(X) have been determined in a line search iteration, conditions of sufficient progress with respect to the objective function are tested. If these conditions are not satisfied, a new value for X is sought in another line search step of interpolation, following a backtracking strategy (i.e., reduction of X2). 

First, when Hk is not positive-definite, the search direction may not exist or may not be a descent direction. Strategies to produce a related positive-definite matrix Hk, or alternative search directions, become necessary. Second, far away from x, the quadratic approximation of expression [34] may be poor, and the Newton direction must be adjusted. A line search, for example, can dampen (scale) the Newton direction when it exists, ensuring sufficient decrease and guaranteeing uniform progress toward a solution. These adjustments lead to the following modified Newton framework (using a line search). 

The strategy described in Section 6.4 for constrained minimization of the sum of squares S 9) is readily adapted to the multiresponse objective function S 9) of Eq. (7.2-16). The quadratic programming subroutine GRQP and the line search subroutine GRS2 are used, with the following changes ... 

The line search procedure in step 3 of Algorithm [Al] is an approximate univariate minimization problem. It is typically performed via quadratic or cubic polynomial interpolation of the one-dimensional function X 0(X) = f k + Xpt). For the polynomial interpolant of ensure that the minimum of is located within the feasible region [xt, x -l- Xpt]. Typically, the initial trial value for X is one. Thus, the line search can be considered as a backtracking algorithm for X in the interval (0, 1]. 

STN Easy search strategy, as translated to Messenger command line language, is visible to the user. Preferences for language of STN Easy screens and on-line help, hit term highlighting, and the number of answers to be displayed can be set by the user. 

Document ordering supports the on-line ordering of original documents the user can select a document supplier. Electronic mail STNmail puts the user in contact with other STN users (e.g., exchange search strategies). 

One important human problem-solving strategy is the application of imagination or intelligent use of a chain of hypotheses to guide the search for an effective line of retrosynthetic analysis. This inductive problem-solving dimension has been discussed previously. 

Sprensen (2001) proposes two strategies that should be initiated as soon as possible (i) a continuous search for breeds or lines that are suitable for organic egg production and (ii) starting a breeding programme for laying hens that takes into consideration the specific requirements of organic egg... 

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