Pattern search method

It may not be possible to develop a mathematical model for the fourth problem it not enough is known to characterize the performance of a rod versus the amounts of the various ingredients used in its manufacture. The rods may have to be manufactured and judged by ranking the rods relative to each other, perhaps based partially or totally on opinions. Pattern search methods have been devised to attack problems in this class. 

The complex method is one such pattern search method (see Fig. 3-54). First, form a complex of at least / -t- 1 (/ = 2 and 4 points are used in Fig. 3-54) different points at which to bake the cakes by picking a range of suitable values for the / independent variables for the baldng process. Bake the cakes and then ask the judges to identify the worst cake. 

Banga et al. [in State of the Art in Global Optimization, C. Floudas and P. Pardalos (eds.), Kluwer, Dordrecht, p. 563 (1996)]. All these methods require only objective function values for unconstrained minimization. Associated with these methods are numerous studies on a wide range of process problems. Moreover, many of these methods include heuristics that prevent premature termination (e.g., directional flexibility in the complex search as well as random restarts and direction generation). To illustrate these methods, Fig. 3-58 illustrates the performance of a pattern search method as well as a random search method on an unconstrained problem. 

FIG. 3-58 Examples of optimization methods without derivatives, (a) Pattern search method. (b) Random search method. O, first phase A, second phase , third phase. 

The Schleyer-Mislow program BIGSTRN (26b) likewise is extensively used. This program incorporates the so-called Engler-Schleyer force field (26a) as well as AUinger s 1971 force field as standard options and is unique in that it uses pattern-search methods of energy minimization. This method is effective at the outset but slows down as the geometry approaches a minimum (Table 2). 

In the actual pattern search method the exploratory moves are made in a way very similar to the univariate search method however, instead of minimizing along the line, we proceed as follows ... 

It is perhaps useful to think of the pattern search method as an attempt to combine the certainty of the multivariate grid method with the ease of the univariate search method, in the sense that it seeks to avoid the enormous numbers of function evaluations inherent in the grid method, without getting involved in the (possibly fruitless and misleading) process of optimizing the variables separately. 

Each of these algorithms was used to adaptively update a two-weight FIR filter and stabilize a Rijke-tube combustor through acoustic actuation. Both the gradient descent and pattern search methods proved quite effective and produced 40 to 50 dB of attenuation of the instability peak. For example, Fig. 18.6 shows the power spectral density of the uncontrolled tube and the system controlled with the Hooke and Jeeves algorithm. The steady-state results of all the... 

An improvement on the simple steepest descent method was made by Schleyer (Engler et al., 1973), and utilized the pattern search procedure. In this case the method proceeds as before, but information regarding the direction of motion of the atoms is saved from one iteration to the next, and the correction terms for succeeding motions are summed, and then applied. The advantage of this method is that if a particular atom is moving down a long hill with small slope, while the steepest descent method requires many iterations to move it, the pattern search method accelerates the motion, because the motion is repeatedly in the same direction, and the size of the correction term increases. 

It is very difficult to compare true efficiencies in minimization algorithms. Depending on the sophistication of the program, the machine system used, and other variables difficult to trace, computation times vary widely, even for the same procedure. It would seem that the pattern search method should be considerably faster than the steepest descent method, other things being equal. 

A practical disadvantage of such a procedure is that one must calculate sufficient elements to fill out a (31V-6) x (3N-6) matrix, which then must be diagonalized. The computing speed for this procedure is a lot faster than for the steepest descent method, and probably is faster than for the pattern search method in a typical case. 

The simplex search algorithm [232-235] is a common pattern search method. A simplex is a geometrical figure that has one more vertex than the dimension of the space in which it operates. That is, n + 1 for n-dimensional systems (3N + 1 for Cartesian coordinates with N atoms) or three vertices (i.e., a triangle) in the two-dimensional case demonstrated in Figure 4.5. 

The objectives of the paper are the maximization of the system availability, the minimization of system cost, and achieving a reasonable ASIL using ASIL-decomposition. In order to make the best choice in the presence of reliahihty, safety, and cost challenges, an objective function and its constraints are defined by authors who involve the above mentioned interests. Subsequently, the optimmn value of objective function will be found using GA and pattern search methods. Adaptability of the optimization results will be finally verified by a practical approach which developed within an automotive project. 

Lewis, Robert Michael and Virginia Torczon, 2000. Pattern Search Methods for Linearly Constrained Minimization, SIAM Journal on Optimization, vol. 10, Number 3, pp. 

Papamichael, E.M. Evmiridis, N.P. (1988). Program based on the pattern search method ... 

Kuriger and Ravindran (2005) have developed three intelligent search methods to solve nonlinear GP problems by adapting and extending the simplex search, complex search, and pattern search methods to account for multiple criteria. These modifications were largely accomplished by using partitioning concepts of GP. The paper also includes computational results with several test problems. 

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