Heuristic method

This family of methods does not require either the gradient g or Hessian G of F. Heuristic methods usually have the foDowrng pros and cons. 

Since these algorithms do not exphcitly require any special function feature, they may be particularly suitable when the function is discontinuous or nonderivable, when the function, gradient or the Hessian are undefined for certain points, and when the gradient or Hessian are computationally onerous. 

Algorithms from this family are less high performance than other alternatives when the function can be approximated by a quadratic function. 

Heuristic methods linearly converge to the solution in the neighborhood of the minimum. 

The simplest method to minimize a multivariable function is based on the random selection of some points in the function domain, hopefully in the neighborhood of the solution. The best point is selected from all the function evaluations. Unfortunately, this approach requires a number of function evaluations that exponentially grows with the problem s dimensions. Hence, it is seldom of interest and, in practice, of no use when the number of variables is even slightly larger than one. 

In order to control a process, it is necessary first to define the desired control reference or setpoint. The setpoint maybe computed as the optimum reference trajectories (u = u°P ), 

Very frequently non-optimal setpoint trajectories are used for controlling reactor temperatures in batch reactors [25,39,179,180]. Reactor temperatures maybe allowed to increase from ambient temperatures up to a maximum temperature value, in order to use the heat released by reaction to heat the reaction medium and save energy (reduce energy costs). The temperature increase is almost always performed linearly, because of hardware limitations and simplicity of controller programming. After reaching the maximum allowed temperature value, reactor temperature is kept constant for a certain time interval, for production of polymer material at isothermal conditions. At the end of the batch, the reaction temperature is increased in order to reduce the residual monomer content of the final resin, usually with the help of a second catalyst. Heuristic optimum temperature trajectories were also formulated for batch polymerizations of acrylamide and quaternary ammonium cationic monomers, in order to use the available heat of reaction [181]. The batch time was split into two batch periods an isothermal reaction period and an adiabatic reaction period. 

A control technique based on high-frequency pressure measurements was developed and implemented to avoid hydrodynamic instabilities in continuous olefin slurry-loop reactors [ 186]. The obtained high-frequency pressure patterns are compared to typical process responses and then used to classify the status of the plant operation. The idea is that pressure fluctuations that do not follow the standard pattern indicate some sort of process instabiUty. When hydrodynamic instabilities are detected, monomer flow rates and/or reactor temperatures are manipulated to reduce the polymer density and the reaction rates and reduce the risks of plant shutdown. Similar procedures can be used for detection and correction of abnormal plant operation in suspension [ 187] and emulsion [188] polymerizations with the help of Raman and near infrared spectroscopy techniques. 

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Also, we consider the total approximation method as a constructive method for creating economical difference schemes for the multidimensional equations of mathematical physics. The notion of additive scheme is introduced as a system of operator difference equations that approximates the original differential equation in the total sense. Two quite general heuristic methods (proposed earlier by the author) for obtaining additive economical schemes are discussed in full details. The additive schemes require a new technique for investigating convergence and a new type of a priori estimates that take into account the definition of the property of approximation. 

Example 7.4-4. Equipment sizing by using the heuristic method (after Sparrow et ai, 1975)... 

The alert reader will object that we are dealing with problems that are unsolvable in full generality and this of course is true. We shall see how often we do indeed end in the "Turing swamp" of undecidable properties. However, the hope is that in applications of interest we can try to skirt the swamp and concentrate on what can be done - and in particular what can. be achieved using heuristic methods or interactive systems. 

In this chapter, we discuss solution approaches for MILP and MINLP that are capable of finding an optimal solution and verify that they have done so. Specifically, we consider branch-and-bound (BB) and outer linearization (OL) methods. BB can be applied to both linear and nonlinear problems, but OL is used for nonlinear problems by solving a sequence of MILPs. Chapter 10 further considers branch-and-bound methods, and also describes heuristic methods, which often find very good solutions but are unable to verify optimality. 

Pekny, J. F. and D. L. Miller. Exact Solution of the No-Wait Flowshop Scheduling Problem with a Comparison to Heuristic Methods. Comput Chem Eng 15(11) 741-748 (1991). 

We believe that this possibility of "guiding" the program is highly didactic since it encourages the student to discover and explore new synthetic routes and to compare the results of "his" strategy with the pre-defined order. That is to say, according to the heuristic method [21] "the pupil is trained to find out things for himself. 

Heuristics are reliable, well-established rules for reducing the number of potential alternative sequences with minimum effort, and often lead to near-optimal separation system designs. Most of the heuristics for distillation sequencing were originally formulated from parametric studies. A number of Heuristics have been suggested, some of which contradict each other (5—8). Heuristic methods have also been extended to sequencing nonsharp distillation separations and to combinations of distillation, mixing, and stream bypass operations (9—11). 

By marrying molecular dynamics to transition state theory, these questionable assumptions can be dispensed with, and one can simulate a relaxation process involving bottlenecks rigorously, assuming only 1) classical mechanics, and 2) local equilibrium within the reactant and product zones separately. For simplicity we will first treat a situation in which there is only one bottleneck, whose location is known. Later, we will consider processes involving many bottlenecks, and will discuss computer-assisted heuristic methods for finding bottlenecks when their locations are not known a priori. 

So far we have considered a system with two reservoirs separated by one bottleneck in general a polyatomic system wil1 have many reservoirs in its configuration space, and the location of the critical bottleneck or bottlenecks will be unknown. Here we will first distinguish critical and rate-limiting bottlenecks from less important ones, and then discuss several more or less heuristic methods for for finding bottlenecks. 

The kinetic temperatures, TK,TR, and T T, are mainly given for completeness. The fact that these three temperatures are not equal indicates that our heuristic method for forcing each system to equilibrate at a preselected temperature was not 100% effective. The largest differences between these temperatures occur in the Stockmayer simulation of CO with p = 1.172 Debyes and in the Lennard-Jones etc., simulation of N2-... 

S. H. Cheng and Y. A. Liu. Studies in chemical process design and synthesis 8. A simple heuristic method for systematic synthesis of initial sequences for sloppy multicomponent separations. I ECRes., 27 2304, 1988. 

Once successfully demonstrated for the original Lifshitz result, the heuristic method can be immediately worked in more complicated geometries. 

In the design of particular separation logic trees, heuristic methods drawing on information related to the physicochemical properties of polypeptides and proteins provide very useful guidelines to facilitate the decision of a preferred option over others for exploration as a subsequent scale-up step in a recovery process. Heuristic methods can be subdivided according to whether the rules are based on the two major constraints of the system. The first of these constraints relates to the separation features of the system per se. Thus,... 

This process assumes advance knowledge of how many basis functions (or data clusters) will be required to appropriately partition the data space. There are numerous heuristic methods of addressing this issue. Since training radial basis function networks is rapid, it is easy to start with a small number of centers and iteratively increase the number until no further benefit is noticed. Note that the dimension of the basis function means is the same as the dimension of the input data vectors. 

Heuristic methods have been widely used for synthesis of distillation sequences to avoid lengthy calculations. Many heuristics are intuitive, e.g., Remove corrosive components first, Remove most plentiful... 

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