Optimization techniques local minima

Another method for finding the minimum is the so-called downhill simplex method [3, 619]. It requires only a function evaluation and does not use either function derivatives or matrix inversitMi. It may be relatively slow if one is trying to optimize many parameters and a shallow minimum, but it will always find a minimum (at least a local minimum). The problem with this technique is that it does not calculate the parameters standard deviations directly. In such cases, it is advisable, after finding the ntinimum by the simplex method, to use these parameters in the CNLS approximation, which should cmiverge quickly and provide standard deviations of the parameters. 

Due to the nrmlinear dependence of the a priori, unknown residual sequence E on the parameter vector 0, the last equation leads to a nonlinear-weighted least squares problem, which has to be tackled by nrmlinear optimization methods. However, nonUnear least squares techniques are sensitive to the initial parameter values and if no acciuate estimates are available, the nuniniization procedure is very likely to converge to a local minimum. In order to avoid potential inaccurate convergence problems associated with arbitrary initial estimates, initial values for the coefficients of projection may be obtained by identifying conventional ARMA models for each of the K data... 

Linear dependence of basis functions, 164 Linear response function, 261 Linear scaling techniques, 80 Linear Synchronous Transit (LST) optimization method, 328 Local Density Approximation (LDA), Local Spin Density Approximation (LSDA) functionals, 182 Local minimum, 339 Localized Molecular Orbitals (LMO), 227 Localized orbital methods, 227 Localized Orbital/local oRiGin (LORG), for calculating magnetic properties, 252 Locally Updated Planes (LUP) optimization method, 330... 

There are also techniques to determine whether we are dealing with a maximum or a minimum, that is, by use of the second derivative. And there are techniques to determine whether we simply have a maximum (one of several local peaks) or the maximum. Such approaches are covered in elementary calculus texts and are well presented relative to optimization in a review by Cooper and Steinberg [2]. 

The multiple-minimum problem is a severe handicap of many large-scale optimization applications. The state of the art today is such that for reasonable small problems (30 variables or less) suitable algorithms exist for finding all local minima for linear and nonlinear functions. For larger problems, however, many trials are generally required to find local minima, and finding the global minimum cannot be ensured. These features have prompted research in conformational-search techniques independent of, or in combination with, minimization.26... 

An optimal control strategy for batch processes using particle swam optimisation (PSO) and stacked neural networks is presented in this paper. Stacked neural networks are used to improve model generalisation capability, as well as provide model prediction confidence bounds. In order to improve the reliability of the calculated optimal control policy, an additional term is introduced in the optimisation objective function to penalise wide model prediction confidence bounds. PSO can cope with multiple local minima and could generally find the global minimum. Application to a simulated fed-batch process demonstrates that the proposed technique is very effective. 

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