Trapped in local minima

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. 

When these methods are unsuitable, nonlinear methods may be applied. The function local minima and overall computational efficiency. The function (u) is often expensive to compute, so maximum advantage must accrue from each evaluation of it. To this end, numerous methods have been developed. Optimization is a field of ongoing research. No one single method is best for all types of problem. Where (u) is a sum of squares, as we have expressed it, and where derivatives dQ>/dvl are available, the method of Marquardt (1963) and its variants are perhaps best. Other methods may be desirable where constraints are to be applied to the vt, or where (u) cannot be formulated as a sum of... 

Backpropagation by gradient-descent is generally a reliable procedure nevertheless, it has its limitations it is not a fast training method and it can be trapped in local minima. To avoid the latter, a variant of the above algorithm called gradient-descent with momentum (GDM) introduces a third term, / ... 

To avoid local minima, most algorithms also test randomly selected points in the error surface. The extent to which a programme carries out these tests determines the speed of locating the minimum and the tendency of the algorithm to become trapped in local minima. 

The picosecond internal dynamics of myoglobin was explored by measuring inelastic neutron scattering by Smith et al. [25]. At low temperatures they found the dynamics to be harmonic while at higher temperatures a considerable quasielastic scattering was detected. Agreement between the experimentally observed spectra and that calculated from molecular dynamics simulations also showed evidence for restriction of the conformational space sampled at 80 K relative to 300 K. On the basis of these results it was concluded that the protein is trapped in local minima at low temperatures in accord with the multiple substate model suggested by low temperature flash photolysis experiments and previous molecular dynamics simulations. Comparison of atomic fluctuation data sets collected at both 325 K and 80 K confirms that the room temperature... 

Conventional gradient base optimisation techniques are not effective to deal with objective functions with multiple local minima and can be trapped in local minima. Particle swam optimisation (PSO) is a recently developed optimisation technique that can cope with multiple local minima. This paper proposes using PSO and stacked neural networks to find the optimal control policy for batch processes. A standard PSO algorithm and three new PSO algorithms with local search were developed. In order to enhance the reliability of the obtained optimal control policy, an additional term is added to the optimisation objective function to penalise wide model prediction confidence bormds. 

The study demonstrates that particle swam optimisation is a powerful optimisation technique, especially when the objective function has several local rninirna. Conventional optimisation techniques could be trapped in local minima but PSO could in general find the global rninimrun. Stacked neural networks can not only given better prediction performance but also provide model prediction confidence bounds. In order to improve the reliability of neural network model based optimisation, an additional term is introduced in the optimisation objective to penalize wide model prediction confidence bormd. The proposed technique is successfully demonstrated on a simulated fed-batch reactor. 

This would indicate a number of frozen states, implying that the system is becoming trapped in local minima. The interconversion times between states would also be expected to be large and grow exponentially with system size.[30-32] In simulated annealing runs, we would expect Xa,b)t to become fixed for T < and large with respect to S Xa,b)t for this case. 

Molecular dynamics calculation is solving Newton s equation of motion using a hybrid target function between empirical force fields and experimental data. The degrees of freedom are the Cartesian coordinates of the atoms. The dynamics aim to cross potential barriers caused by inappropriately folded structures. This can reduce the problem of being trapped in local minima more than the torsion angle space minimization. Usually, a... 

The graphics Interaction section of MOLMEC contains routines capable of rotating and aligning the molecule into any desired position. Since the graphics unit is a two-dimensional screen, rotation is essential to obtain a good view of the structure. Furthermore, these routines are useful in locating atoms trapped in local minima. If such an atom Is found, the user can move the trapped atom to a new position by a MOVE routine found in the graphics section. 

Samples such as this three-component mixture are difficult to analyze. In this case we know that there are three decay times, and, in fact, the three decay times are correctly determined by the analysis. However, obtaining the correct values required that the parameter starting values in the least-squares analysis be close to the correct values. Otherwise, the program stopped at incorrect values, apparently trapped in local minima. Additionally, the Xa surface is essentially independent of lifetime, as shown in Figure 5.21 for the data measured at 380 nm (dashed line). Hence, without prior knowledge of the presence of three decay times, one would not know whether to accept the two- or three-decay-time fit. 

Note that for small errors, Eq. (19.31) converges to the derivative of activation function at the point of the output value. With an increase of system dimensionality, the chances for local minima decrease. It is believed that the described phenomenon, rather than a trapping in local minima, is responsible for convergency problems in the error backpropagation algorithm. 

Notice that the function exhibits many local minima besides the global minimum located at (0,0). The PSO parameters are as follows Ci = C2 = 1.5 and x = 0.7. The inertia parameter w was linearly adjusted from 1.5 to 0.5 along K = 25 steps. To do this use was made of iV = 10 particles. The right panel of the figure shows that the global minimum was successfully found by the algorithm without being trapped in local minima. 

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