Optimization stop criteria

Cronk, a., and Macon, M. Optimized stopping criteria for tree-based unit selection in concatenative S3mthesis. In Proceedings of the International Conference on Speech and Language Processing 1998 (1998). 

After an initial starting geometry has been generated and optimized (e.g., in a force field), the new conformation is compared with all the previously generated conformations, which are usually stored as a list of unique conformations. If a substantially different geometry is detected it is added to the list otherwise, it is rejected. Then a new initial structure is generated for the next iteration. Finally, a preset stop criterion, e.g., that a given number of loops has been performed or that no new conformations can be found, terminates the procedure. 

No single stopping criterion will suffice for Newton s method or any of the optimization methods described in this chapter. The following simultaneous criteria are recommended to avoid scaling problems ... 

The Hamiltonian, the adjoint equations and the optimal reflux ratio correlation will be same as those in Equations P.10-P.13 (Diwekar, 1992). However, note that the final conditions (stopping criteria) for the minimum time and the maximum distillate problems are different. The stopping criterion for the minimum time problem is when (D, xq) is achieved, while the stopping criterion for the maximum distillate problem is when t, xo) is achieved. See Coward (1967) for an example problem. 

Confidence intervals may be used to define a stop criterion, i.e. they can be used to judge whether the optimization process should be continued or halted. If the predicted optimum falls within one of the existing confidence intervals (calculated for 5=0.025), then the experimental capacity factors will be within 2.5% of the predicted values. It should be noted that an error of 2.5% in k can make a big difference if the relative retention (a) of a pair of peaks is close to one. It may therefore be required to use a lower value for S in eqn.(5.19). 

It may not be possible to achieve sufficient resolution for all the pairs of peaks in the chromatogram on the particular column. Therefore, a stop criterion is needed in the optimization procedure, for instance a maximum of two attempts to separate a particular pair of peaks. If it is difficult to recognize (pairs of) peaks, then a maximum of two or three optimization cycles each for stage 2 and stage 3 of the optimization procedure may be considered. 

The convergence criterion in the alternating least-squares optimization is based on the comparison of the fit obtained in two consecutive iterations. When the relative difference in fit is below a threshold value, the optimization is finished. Sometimes a maximum number of iterative cycles is used as the stop criterion. This method is very flexible and can be adapted to very diverse real examples, as shown in Section 11.7. 

Obtaining the solution requires 14770 generations (295421 goal function evaluations) for a stop criterion of 6000 generations. The optimal sensor network is obtained after 301 seconds on a 1.6GHz computer. This optimal network costs 1860 units and counts 25 sensors, one for each possible sensor location. It allows detecting and locating all the 15 faults. The initial and most expensive network costs 3100 units (1240 cost units more than the optimal one). 

Mitra et al. (1998) employed NSGA (Srinivas and Deb, 1994) to optimize the operation of an industrial nylon 6 semibatch reactor. The two objectives considered in this study were the minimization of the total reaction time and the concentration of the undesirable cyclic dimer in the polymer produced. The problem involves two equality constraints one to ensure a desired degree of polymerization in the product and the other, to ensure a desired value of the monomer conversion. The former was handled using a penalty function approach whereas the latter was used as a stopping criterion for the integration of the model equations. The decision variables were the vapor release rate history from the semibatch reactor and the jacket fluid temperature. It is important to note that the former variable is a function of time. Therefore, to encode it properly as a sequence of variables, the continuous rate history was discretized into several equally-spaced time points, with the first of these selected randomly between the two (original) bounds, and the rest selected randomly over smaller bounds around the previous generated value (so as... 

The most important stopping criterion is the satisfaction of the DM in some solution. (Some interactive methods use also algorithmic stopping criteria but we do not go into such details here.) In each iteration, some information about the problem or solutions available is given to the DM and then (s)he is supposed to answer some questions or to give some other kind of information. New solutions are generated based on the information specified. In this way, the DM directs the solution process towards such Pareto optimal solutions that (s)he is interested in and only such solutions are generated. 

If some stopping criterion has been reached, then the algorithm proceeds to Step 10 where the Lawton matrix condition is verified. Provided Conditions 1 and 2 of Section 4 hold, then the Lawton matrix condition will not be satisfied for exceptional degenerate cases, thus the Lawton matrix is verified after the adaptive wavelet has been found. Finally, the multivariate statistical procedure can be performed using the coefficients X " (to). The optimizer used in the adaptive wavelet algorithm is the default unconstrained MAT-LAB optimizer [12]. 

The stopping criterion depends on the specific properties to be fitted. For example, if the density deviates by less than 0.5 % from experiment, the corresponding force field is considered as optimal because the experiment is not more accurate either. The same holds for all other properties. However, the experimental accuracy is much lower for transport properties like diffusion coefficients or viscosity. 

A majority of the modules requires certain input parameters, which have to be defined in a user-written configuration file, and is read by the main python module main.py. The configuration file specifies all class objects, modules, and submodules that are desired for optimization process. It also contains important preferences concerning the system (e.g., inpul/output paths, number of computer cores, batch system), the optimization (e.g., algorithm, step length control, stopping criterion, initial parameters, constraints), and the optimization problem (e.g., objective functions, the loss function s target values). When molecular simulations are performed, all desired properties and parameters of the thermodynamic system have to be defined (e.g., ensemble, temperatures, pressures, physical properties to be fitted, number of molecules, box size, number of MD/MC steps, time step). Hence, the file is divided into three blocks. If more than one substance is considered in the optimization, one block for each substance has to be indicated. 

The following procedure is adopted as the stop criterion. At each iteration, the number of points in the neighborhood of the optimum (given a tolerance value) is checked. If such a number is reasonable (according to an assigned value), a possible solution is reached. Theoretically, the number of points should be in the order of magnitude of the optimization problem dimensions, but it is preferable to use smaller numbers when the optimization size is large. 

In an earlier section, we had alluded to the need to stop the reasoning process at some point. The operationality criterion is the formal statement of that need. In most problems we have some understanding of what properties are easy to determine. For example, a property such as the processing time of a batch is normally given to us and hence is determined by a simple database lookup. The optimal solution to a nonlinear program, on the other hand, is not a simple property, and hence we might look for a simpler explanation of why two solutions have equal objective function values. In the case of our branch-and-bound problem, the operationality criterion imposes two requirements ... 

Iterative resolution methods obtain the resolved concentration and response matrices through the one-at-a-time refinement or simultaneous refinement of the profiles in C, in ST, or in both matrices at each cycle of the optimization process. The profiles in C or ST are tailored according to the chemical properties and the mathematical features of each particular data set. The iterative process stops when a convergence criterion (e.g., a preset number of iterative cycles is exceeded or the lack of fit goes below a certain value) is fulfilled [21, 42, 47-50],... 

Stopping because criteria (2-4) are met is considered desirable and is in consequence called optimal in the output. Stop by criterion (5) is called fiasco", and the actual value of X is printed, with a proposal to increase it. ... 

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