Local optimization methods

For molecular systems in the vacuum, exact analytical derivatives of the total energy with respect to the nuclear coordinates are available [22] and lead to very efficient local optimization methods [23], The situation is more involved for solvated systems modelled within the implicit solvent framework. The total energy indeed contains reaction field contributions of the form ER(p,p ), which are not calculated analytically, but are replaced by numerical approximations Efp(p,p ), as described in Section 1.2.5. We assume from now on that both the interface Y and the charge distributions p and p depend on n real parameters (A, , A ). In the geometry optimization problem, the A, are the cartesian coordinates of the nuclei. There are several nonequivalent ways to construct approximations of the derivatives of the reaction field energy with respect to the parameters (A1 , A ) ... 

Another approach is known as the local optimization method. Here local means that maximization of the objective function J is carried out at each time, i.e., locally in time between 0 and tf. There are several methods for deriving an expression for the optimal laser pulse by local optimization. One is to use the Ricatti expression for a linear time-invariant system in which a differential equation of a function connecting [r(t) and (f) is solved, instead of directly solving for these two functions. Another method... 

Local optimization methods have experienced far more extensive development in the last decade. Studies have produced a range of robust and reliable... 

Another strategy to circumvent to be trapped in a local optimum is simulated annealing (SA). Annealing is a process in which a solid material is first melted and then allowed to cool by slowly decreasing the temperature. In contrast to conventional, local optimization methods, in SA steps are allowed in a direction that yields inferior solutions. To control these steps, a probability function that originates from Metropolis work in statistical thermodynamics is exploited. 

In analogy to a typical local optimization method, such as the simplex method (cf. Section 4.3), SA could be programmed as given in Example 8.10. 

The temperature and cooling parameter a have to be chosen carefully. Too high a temperature leads to acceptance of almost all inferior solutions and the search might tend toward a random search. If the temperature is too low, almost no inferior solution will be accepted and the search approaches one of the local optimization methods. Commonly, the probability for acceptance of an inferior solution should be higher than 0.5. If no information on the temperature is at hand, the algorithm can be initiated at a lower temperature that is gradually increased until an acceptable probability of inferior solutions is found. Then the coohng process... 

Compare the optimization strategies of genetic algorithms, simulated annealing, and tabu search with those of local optimization methods. 

Wienke et al. 27 compared the GA with several standard optimization techniques including simulated annealing, grid search, simplex, pattern search, along with local optimization methods, for several test problems. Their conclusion was that the GA consistently outperformed the other methods as measured by the fraction of runs that found the global optimum. 

The multi-structure methods are quite tolerant towards the presence of many soft degrees of freedom, which often causes problems with the local optimization methods described in Sections 12.4.5-12.4.7. Multi-structure methods such as NEB are therefore well suited for large systems such as extended (periodic) systems. 

Pekkanen M. (1995). A local optimization method for the design of reactive distillation. Computers and Chemical Engineering 19 (SS), S235-S240. 3.3, 3.3.1... 

The current decision process of the manufacturer, which wire harness shall be ordered, is based on a local optimization method. Because of the fact, that the task is a discrete optimization problem, a local optimization can differ significantly from the global optimum (see Chap. 15). This requires a further step for additional optimization. 

In contrast to the SGO and SLO methods, the temporal local optimization method (TLO) assumes that the optical flow field is constant temporally (Fig. 6) [44] ... 

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