Powell s method

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. 

A modification of Powell s method (P4), with sodium benzoate and urea solution to speed up the coupling reaction, has been used by O Hagan (01) to investigate the bilirubin content of serum from a large number of blood donors. He stated that color development was maximal in half a minute and that errors due to hemoglobin and ferrihemalbumin could be eliminated by the use of suitable blanks. The mean values for serum bilirubin in 200 donors were 0.45 0.19 mg/100 ml serum for males and 0.36 0.18 mg/100 ml serum for females. Of 2500 people screened, only 108 had unaccountably high levels of over 1.5 mg j0, and this level was arbitrarily taken as the upper limit of normal. 

George, et al. (28) implemented Powell s method (29), a quasi-Newton method, that uses approximations to the Hessian matrix and its inverse, H u) and H u l, to calculate new compositions. 

A FORTRAN program is available from the authors. For each system, uj are assigned for species in phases expected at equilibrium. As uj become small for all species in a phase, variables are retained and new values computed using Powell s method. 

The situation with respect to Powell s method is therefore far from clear and, in the absence of any systematic work to determine its effectiveness in a quantum chemical context, it seems to the Reporters that the method is always worth a preliminary try in an optimization problem particularly as it is so easy to program and so compact. 

Nonderivative methods include random search, grid search, simplex search, and conjugate directions (or Powell s method). The nonderivative methods use various patterns for generating new test points for decision variables, and then a comparison of the new objective function value against previous values. A subsequent test point is then generated, either based on the immediate comparison or using the previous history of test points. 

One method to finding the minimum of a function is to search over all possible values of x and find the minimum based on the corresponding values of Y. These algorithms include region elimination methods, such as the Golden Search algorithm or point-estimation methods, such as Powell s method, neither of which require calculation of the derivatives. However, these types of methods often assume unimodality, and in some cases, continuity of the function. The reader is referred to Reklaitis Ravindran, and Ragsdell (1983) or Rao (1996) for details. 

Direction set (Powell s) methods belong to a class of local optimisation methods in multidimensions. The starting point x(p, ..., p ) in n-dimen-sional space is evolved in some vector direction v so that the function F(x) is minimised along the line v. The method proceeds as follows ... 

One valid alternative to the Levenberg-Marquardt method is the dogleg method, also known as Powell s hybrid method (Rabinowitz, 1970). Once again, this couples the Newton and gradient methods. The original version of Powell s method was close to the tmst region concept. Powell proposed a strategy for the modification of parameter d subject to both the successes and failures of the procedure. 

An efficient and robust optimisation algorithm is primordial for this solution strategy. Rao Sawyer (1995) applied Powell s method to tackle the optimisation. Koyliioglu et al. (1995) defined a linear programming solution for this purpose. The input interval vector defines the number of constraints and, therefore, strongly influences the performance of the procedure. Also, because of the required execution of the deterministic FE analysis in each goal function evaluation, the optimisation approach is numerically expensive. Therefore, this approach is best suited for rather small FE models with a limited number of input uncertainties, unless approximate methods can be used that avoid the expensive iterative calculation of the entire FE system of equations. 

The combination of successive substitution and Newton s method is a good choice and has the desirable features of both. In this approach, the successive substitution comprises the first few iterations and later, when a switching criterion is met, Newton s method is used. To our knowledge, some commercial reservoir simulation models have adopted the combined successive substitution-New ton approach after the experience with various methods of solving nonlinear flash calculation including Powell s method (1970). The application of a reduction method to phase equilibrium calculations has also been proposed (Michelsen, 1986 Hendriks, and Van Bergen, 1992). In this approach, the dimensionality of phase equilibrium problems for multicomponent mixtures can be drastically reduced. The application of reduction methods and its implementation in reservoir compositional models is under evaluation. 

This equation can be solved numerically in three different ways. The first is by direct integration of equation (8.21) and is described in Section 8.5.2. The second method is to treat the objective function as an unconstrained optimization, which can be solved using any direct search technique such as Powell s method (1964). This approach is computationally faster than the direct integration but requires good initial estimates for the volumes. The third method is to differentiate equation (8.21) to give two nonlinear equations ... 

---