Optimization unimodal

Pore size optimization is one area where developmental efforts have been focused. Unimodal pore (NiMo) catalysts were found highly active for asphaltene conversion from resids but a large formation of coke-like sediments. Meanwhile, a macroporous catalyst showed lower activity but almost no sediments. The decrease of pore size increases the molecular weight of the asphaltenes in the hydrocracked product. An effective catalyst for VR is that for which average pores size and pore size distribution, and active phase distribution have been optimized. Therefore, the pore size distribution must be wide and contain predominantly meso-pores, but along with some micro- and macro-pores. However, the asphaltene conversion phase has to be localized in the larger pores to avoid sediment formation [134],... 

Optimization problems in crystallographic structure refinement are seldom convex, that is very rarely characterized by a unimodal function/(x). Regularization of a two-atom model is an example of such a unimodal function. Fig. 11.2a. in contrast. Fig. 11.2b shows a profile of a function for modelling an amino acid side chain - the peaks correspond to the possible rotamers. In this case, the shape of the function/(x) is called multimodal. Such functions arise naturally in structural macromolecular optimization problems and possess a highly complex multiminima energy landscape that does not lend itself favourably to standard robust optimization techniques. 

Unconstrained optimization deals with situations where the constraints can be eliminated from the problem by substitution directly into the objective function. Many optimization techniques rely on the solution of unconstrained subproblems. The concepts of convexity and concavity will be introduced in this subsection, as well as discussing unimodal versus multimodal functions, singlevariable optimization techniques, and examining multi-variable techniques. 

For the optimization situation in which two or more independent variables are involved, response surfaces can often be prepared to show the relationship among the variables. Figure 11-12 is an example of a unimodal response surface with a single minimum point. Many methods have been proposed for exploring such response surfaces to determine optimum conditions. 

In our study of search problems we have seen that single variable systems can be optimized with ease two-variable systems, with some effort and multivariable systems, only with extreme difficulty if at all. As more variables enter a search problem, the number of experiments needed grows rapidly, and the unimodality assumption becomes less and less plausible. Thus our investigation of search problems leads directly to interaction problems, where the criterion of effectiveness depends on so many factors that it is impractical, or even impossible, to find the optimum by conventional methods. Successful techniques for solving interaction problems involve decomposing a big system into several smaller ones, as we have already done with our lines of search. 

To deduce a particle size distribution, the detector response must be deconvoluted by means of a simulation calculation. The scattering particles are assumed to be spherical in shape, and the data are subjected to one of three different computational methods. One system uses the unimodal model-dependent method, which begins with the assumption of a model (such as log normal) for the size distribution. The detector response expected for this distribution is simulated, and then the model parameters are optimized by minimizing the sum of squared deviations from the measured and the simulated detector responses. The model parameters are finally used to modify the originally chosen size distribution, and it is this modified distribution that is presented to the analyst as the final result. 

A second approach uses the unimodal model-independent method, which begins with the assumption that the size distribution consists of a finite number of fixed size classes. The detector response expected for this distribution is simulated, and then the weight fractions in each size class are optimized through a minimization of the sum of squared deviations from the measured and simulated detector responses. The third system uses the multimodal model-independent method. For this, diffraction patterns for known size distributions are simulated, random noise is superimposed on the patterns, and then the expected element responses for the detector configuration are calculated. The patterns are inverted by the same minimization algorithm, and these inverted patterns are compared with known distributions to check for qualitative correctness. 

In principle, optimal wx and pa can be determined by minimizing the summed variance of the Monte Carlo integrations used to evaluate Eq. (2.8), but this would be even more complicated than minimizing the variance in the unimodal case. Instead we use a two-step approximation procedure. 

If the function to be optimized is unimodal, an object from the BzzM In Im 1-zatlonMono class may be used (see Vol. 3, Buzzi-Ferraris and Manenti, 2014). 

When one optimal solution exists in the feasible region, the objective function is uni-modal. When two optimal solutions exist, it is bimodal if more than two, it is multimodal. LP problems are imimodal unless the constraints are inconsistent, such that no feasible region exists. The solutions in Figures 18.1 to 18.3 are unimodal. A two-dimensional, multimodal case is shown in Figure 18.4, taken from Reklaitis et al. (1983) and called the Himmelblau problem. This is an unconstrained problem with the objective function ... 

The golden-section search method determines the optimal solution to a bounded objective function that is one-dimensional and unimodal. However, the function need not be continu-... 

If the function to be optimized is unimodal, an object from the BzzMinimiza-tionMono class may be used. This class can be combined with classes used for the integration of differential systems based on multivalue algorithms (see Vol. 4 -Buzzi-Eerraris and Manenti, in press). The objects from these classes automatically change the integration step and the order of the algorithm as well by adapting them to the problem s features and certain specific requirements. 

---