Algorithms Simplex

LP software includes two related but fundamentally different kinds of programs. The first is solver software, which takes data specifying an LP or MILP as input, solves it, and returns the results. Solver software may contain one or more algorithms (simplex and interior point LP solvers and branch-and-bound methods for MILPs, which call an LP solver many times). Some LP solvers also include facilities for solving some types of nonlinear problems, usually quadratic programming problems (quadratic objective function, linear constraints see Section 8.3), or separable nonlinear problems, in which the objective or some constraint functions are a sum of nonlinear functions, each of a single variable, such as... 

Fig. 5.4 The three basic moves permitted to the simplex algorithm (reflection, and its close relation reflect-and-expmd contract in one dimension and contract around the lowest point). (Figure adapted from Press W H, B P Flannery,...

Fig. 5.5 The first few steps of the simplex algorithm with the function + 2i/. The initial simplex corresponds to the triangle 123. Point 2 has the largest value of the function and the next simplex is the triangle 134. The simplex for tire third step is 145.

Because the size of the simplex remains constant during the search, this algorithm is called a fixed-sized simplex optimization. Example 14.1 illustrates the application of these rules. 

Find the optimum response for the response surface in Figure 14.7 using the fixed-sized simplex searching algorithm. Use (0, 0) for the initial factor levels, and set the step size for each factor to 1.0. 

The relationship between output variables, called the response, and the input variables is called the response function and is associated with a response surface. When the precise mathematical model of the response surface is not known, it is still possible to use sequential procedures to optimize the system. One of the most popular algorithms for this purpose is the simplex method and its many variations (63,64). 

The simplex algorithm, in a sense, prepares the problem before cal-... 

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. 

To determine the optimal parameters, traditional methods, such as conjugate gradient and simplex are often not adequate, because they tend to get trapped in local minima. To overcome this difficulty, higher-order methods, such as the genetic algorithm (GA) can be employed [31,32]. The GA is a general purpose functional minimization procedure that requires as input an evaluation, or test function to express how well a particular laser pulse achieves the target. Tests have shown that several thousand evaluations of the test function may be required to determine the parameters of the optimal fields [17]. This presents no difficulty in the simple, pure-state model discussed above. 

Three paths can be advanced (1) expansion, e.g., Taylor series (2) trial and error, e.g., generating curves on the plotter and (3) simplex optimization algorithm. (See Section 3.1.)... 

Purpose To determine, from eight initial experiments performed under certain conditions, whether the three controlled parameters have an effect on the measurement, and which model is to be used. This factorial approach to optimization is an alternative to the use of multidimensional simplex algorithms it has the advantage of remaining transparent to the user. 

A family of computer programs has been rritten for this procedure called MIXCO. The algorithm used for the iteration and optimization processes is the simplex routine and was adapted... 

The selection to minimize absolute error [Eq. (6)] calls for optimization algorithms different from those of the standard least-squares problem. Both problems have simple and extensively documented solutions. A slight advantage of the LP solution is that it does not need to be solved for the points for which the approximation error is less than the selected error threshold. In contrast, the least squares problem has to be solved with every newly acquired piece of data. The LP problem can effectively be solved with the dual simplex algorithm, which allows the solution to proceed recursively with the gradual introduction of constraints corresponding to the new data points. 

To perform the maximization over (X,t), we need an algorithm such as the Nelder-Mead simplex search (14). An alternative that is adequate in many cases is a simple search over a (X,t) grid. The critical value XX has an interpretation of its own. It is the upper bound on a simultaneous prediction interval for ng as yet unobserved observations from the background population. 

Basically two search procedures for non-linear parameter estimation applications apply. (Nash and Walker-Smith, 1987). The first of these is derived from Newton s gradient method and numerous improvements on this method have been developed. The second method uses direct search techniques, one of which, the Nelder-Mead search algorithm, is derived from a simplex-like approach. Many of these methods are part of important mathematical computer-based program packages (e.g., IMSL, BMDP, MATLAB) or are available through other important mathematical program packages (e.g., IMSL). 

The Simplex algorithm and that of Powell s are examples of derivative-free methods (Edgar and Himmelblau, 1988 Seber and Wild, 1989, Powell, 1965). In this chapter only two algorithms will be presented (1) the LJ optimization procedure and (2) the simplex method. The well known golden section and Fibonacci methods for minimizing a function along a line will not be presented. Kowalik and Osborne (1968) and Press et al. (1992) among others discuss these methods in detail. 

The technique is useful where the problem is to decide the optimum utilisation of resources. Many oil companies use linear programming to determine the optimum schedule of products to be produced from the crude oils available. Algorithms have been developed for the efficient solution of linear programming problems and the SIMPLEX algorithm, Dantzig (1963), is the most commonly used. 

To offer more flexibility we adopt an approach, based on the transient simulation model TRNSYS (Klein et al., 1976), making use of the Lund DST borehole model (Hellstrom, 1989). The parameter estimation procedure is carried out using the GenOPT (Wetter, 2004) package with the Nelder and Mead Simplex minimization algorithm (Nelder and Mead, 1965) or Hooke and Jeeves minimization algorithm (Hooke and Jeeves, 1961). 

Figure 53. Example of GenOpt calibration run, using a Nelder Mead Simplex algorithm...

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