Complete mathematical optimization

A second reason not to become involved in extensive calculations for the complete mathematical optimization of the (primary) program parameters is that a more powerful way to optimize the separation of all sample components in the mixture may be to optimize the selectivity of the gradient by varying the nature of the mobile phase components (secondary parameters). 

Simplex method Snyder method Jandera method Complete mathematical optimization... 

This technique requires that the experimentation be completed before optimization so that mathematical... 

In the mathematical optimization based approaches first a superstructure is created which has embedded a large number of alternative designs. Then mathematical techniques like MINLP are used to find the optimum process within the specified superstructure. For the products considered here there are two big hurdles preventing the large scale use of these techniques (Hill, 2004). Firstly a lot of the physico-chemical phenomena occurring are not completely understood. This makes rigorous modeling difficult. Secondly there is a lack of relevant property models for structured products. 

The reader will find here a complete mathematical development of the models of chromatography and other physical laws which direct the chemical engineer in the design and scale-up of chromatographic processes. For preparative chromatographic separations, our ultimate purpose is the optimization of the experimental conditions for maximum production rate, minimum solvent consumption, or minimum production cost, with or without constraints on the recovery yield. The considerable amormt of work done on this critical topic is presented in the... 

The proposed methodology for computer-aided optimal design in the development of YSZ-based gas sensors comprises three phases. Firstly, the complete mathematical model with distributed temporal and spatial parameters for electrochemical gas sensors is presented as a system of the differential equations in private derivatives of parabolic and hyperbolic types. The complexity of physical and chemical interactions, represented in this model, allows performing a mathematical description of the electrochemical gas sensors toward standardization of the calculating procedures. The complete mathematical model and the algorithm of transfer from the complete... 

By using the deduction principle, the complete mathematical model of the electrochemical gas sensors with distributed parameters can be transformed to the mathematical models of the specific gas sensors, which is important for organization of their optimal design. 

A complete mathematical programming formulation of the problem requires specification of an objective function that has to be optimized and a set of constraints that have to be satisfied. The objective function can be whatever the designer wishes e.g., the total system cost, or the overall... 

The chromatographic system should, wherever possible, be optimized to obtain complete resolution of the mixture and not place reliance on mathematical techniques to aid in the analysis. 

Any numerical experiment is not a one-time calculation by standard formulas. First and foremost, it is the computation of a number of possibilities for various mathematical models. For instance, it is required to find the optimal conditions for a chemical process, that is, the conditions under which the reaction is completed most rapidly. A solution of this problem depends on a number of parameters (for instance, temperature, pressure, composition of the reacting mixture, etc.). In order to find the optimal workable conditions, it is necessary to carry out computations for different values of those parameters, thereby exhausting all possibilities. Of course, some situations exist in which an algorithm is to be used only several times or even once. 

The techniques most widely used for optimization may be divided into two general categories one in which experimentation continues as the optimization study proceeds, and another in which the experimentation is completed before the optimization takes place. The first type is represented by evolutionary operations and the simplex method, and the second by the more classic mathematical and search methods. (Each of these is discussed in Sec. V.)... 

Rather than find the most perfect MOs which satisfy eqn (10-2.4) (or eqns (10-2.5)), it is common practice to replace them by particular mathematical functions of a restricted nature. These functions will generally contain certain parameters which can then be optimized in accordance with eqn (10-2.4). Since these MOs are not completely flexible, we will have introduced a further approximation, the severity of which is determined by the degree of inflexibility in the form of our chosen functions. Typical of this kind of approximation is the one which expresses the space part of the MOs as various linear combinations of atomic orbitals centred on the same or different nuclei in the molecule. We write the space part of each of the approximate MOs as... 

Modern computational chemistry programs usually have various well-tested optimization algorithms for finding the lowest energy (33). Such algorithms have adjustable mathematical parameters that were chosen by the program developer to balance between computer time requirements and completeness of the minimization. It has been pointed out that reliable... 

Finally, it should be pointed out once again that obtaining as precise and complete information on a studied chemical or physical system as possible, with a minimal number of experiments and the lowest possible expenses, is the necessary condition for efficient research work. Therefore, application of modern mathematical and statistical methods in designing and analyzing experimental results is a real necessity in all fields and phases of work, starting with purely theoretical considerations of a process, its research and development, all the way to designing equipment and studying optimal operational conditions of a plant. 

The last twenty years of the last millennium are characterized by complex automatization of industrial plants. Complex automatization of industrial plants means a switch to factories, automatons, robots and self adaptive optimization systems. The mentioned processes can be intensified by introducing mathematical methods into all physical and chemical processes. By being acquainted with the mathematical model of a process it is possible to control it, maintain it at an optimal level, provide maximal yield of the product, and obtain the product at a minimal cost. Statistical methods in mathematical modeling of a process should not be opposed to traditional theoretical methods of complete theoretical studies of a phenomenon. The higher the theoretical level of knowledge the more efficient is the application of statistical methods like design of experiment (DOE). 

If the behaviour of complex chemical (in our case catalytic) reactions is known, it will be clear in what way these reactions can be carried out under optimal conditions. The results of studying kinetic models must be used as a basis for the mathematical modelling of chemical reactors to perform processes with probable non trivial kinetic behaviour. It is real systems that can appear to show such behaviour first far from equilibrium, second nonlinear, and third multi dimensional. One can hardly believe that their associated difficulties will be overcome completely, but it is necessary to approach an effective theory accounting for several important problems and first of all provide fundamentals to interpret the dependence between the type of observed kinetic relationships and the mechanism structure. 

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