Search Directions

A con jugate gradicri I method differs from the steepest descent technique by using both the current gradient and the previous search direction to drive the rn in im i/ation. , A conjugate gradient method is a first order in in im i/er. 

The advan tage ol a conjugate gradien t m iniim/er is that it uses th e minim i/ation history to calculate the search direction, and converges t asLer Lhan the steepest descent technique. It also contains a scaling factor, b, for determining step si/e. This makes the step si/es optimal when compared to the steepest descent lechniciue. 

Another difference from steepest descent is that a one-diinen-sional minimization is performed in each search direction. Aline mmimi/ation is made along a direction h until a minlmnni energy is found at anew point i-i-l then the search direction is updated and a search down the new direction h ] is made. This... 

The conjugate direction is reset to the steepest descent direction every 3N search direction s or cycles, or if the en ergy rises between cycles. 

Another difference from steepest descent is that a one-dimensional minimization is performed in each search direction. Aline minimization is made along a direction hj until a minimum... 

Finally, to ensure convergence of this algorithm from poor starting points, a step size Ot is chosen along the search direction so that the point at the next iteration z = z- + Ctd) is closer to the solution of the... 

There are two basic types of unconstrained optimization algorithms (I) those reqmring function derivatives and (2) those that do not. The nonderivative methods are of interest in optimization applications because these methods can be readily adapted to the case in which experiments are carried out directly on the process. In such cases, an ac tual process measurement (such as yield) can be the objec tive function, and no mathematical model for the process is required. Methods that do not reqmre derivatives are called direc t methods and include sequential simplex (Nelder-Meade) and Powell s method. The sequential simplex method is quite satisfac tory for optimization with two or three independent variables, is simple to understand, and is fairly easy to execute. Powell s method is more efficient than the simplex method and is based on the concept of conjugate search directions. 

The steepest descent method is quite old and utilizes the intuitive concept of moving in the direction where the objective function changes the most. However, it is clearly not as efficient as the other three. Conjugate gradient utilizes only first-derivative information, as does steepest descent, but generates improved search directions. Newton s method requires second derivative information but is veiy efficient, while quasi-Newton retains most of the benefits of Newton s method but utilizes only first derivative information. All of these techniques are also used with constrained optimization. 

The conjugate gradient method is one of the oldest in the Retcher-Reeves approach, the search direction is given by... 

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... 

The penalty function for the laser fiuence, which gives rise to the term involving (3 in Eq. (10), acts to limit the magnitude of the electric field to within physically acceptable limits. In order to further limit the field strength the search direction is projected as follows [101] ... 

Straightforward application of OCT as described above often results in a quite complicated pulse shapes and may especially introduce some high frequency components, which are difficult to realize experimentally, into the pulse. It is thus highly desirable to find an optimized pulse with spectral components within a predefined frequency range. With this end in view the projected search direction is subjected to a spectral filter... 

Here (j is the CG update parameter. In the above equations, e = e (tj) o vector notation for the discretized electric field strength, = g (fj) o objective functional J with respect to the field strength (evaluated at a field strength of e t) and dk = d (t ) o search direction at the feth iteration. The time has been discretized into N time steps, such as that tj=jx )t, where j = 0,1,2, , N. Different CG methods correspond to different choices for the scalar (j. ... 

Nowadays, MS is often no longer the analytical bottleneck, but rather what precedes it (sample preparation) and follows it (data handling, searching). Direct mass-spectrometric methods have to compete with the separation techniques such as GC, HPLC and SFC that are commonly used for quantitative analysis of polymer additives. Extract analysis has the general advantage that higher-molecular-weight (less-volatile) additives can be detected more readily than by direct analysis of the polymer compound. 

The method of steepest descent uses only first-order derivatives to determine the search direction. Alternatively, Newton s method for single-variable optimization can be adapted to carry out multivariable optimization, taking advantage of both first- and second-order derivatives to obtain better search directions1. However, second-order derivatives must be evaluated, either analytically or numerically, and multimodal functions can make the method unstable. Therefore, while this method is potentially very powerful, it also has some practical difficulties. 

Initial search is in the direction of steepest descent given by the reduced gradient, z0 say. Subsequent search directions sk+, are generated by a conjugate direction formula (F6),... 

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