Classification of Optimization Problems

Inequality constraints, g x, are expressions that involve any or all of the set of variables, i, and are used to bound the feasible region of operation. For example, when operating a centrifugal pump, the head developed decreases with increasing flow rate according to a pump characteristic curve. Hence, if the flow rate is varied when optimizing the process, care must be taken to make sure that the required pressure increase (head) does not exceed that available from the pump. The expression might be of the form. 

Similar kinds of constraints involve the reflux ratio in distillation, which must exceed the minimum value for the required separation. If the distillation tower pressure is adjusted, the minimum reflux ratio will change and the actual ratio must be maintained above the minimum value. Even when optimization is not performed, the decision variable values must be selected to avoid violating the inequality constraints. In some cases, the violations can be detected when examining the simulation results. In other cases, the imit subroutines are unable to solve the equations as, for example, when the reflux ratio is adjusted to a value below the minimum value for a specified split of the key components. 

Some inequality constraints simply place lower and upper bounds, x on any or all of the variables, x. Others permit the specification of just a lower bound or just an upper bound, for example, a lower bound on the fractional recovery of a species in a product stream. Sometimes the upper and lower bounds are included with the inequality constraints, but here, they are considered separately. 

The combination of the equality constraints, inequality constraints, and lower and upper bounds defines a feasible region. A feasible solution is one that satisfies the equality constraints, the inequality constraints, and the upper and lower bounds for a feasible set of decision variables. If the solution also minimizes (or maximizes) the objective function, it is a local optimal solution. Sometimes, other local optimal solutions exist in the feasible region, with one or more being a global optimal solution. 

Many numerical methods have been devised for solving optimization problems. The choice of method depends upon the nature of the formulation of the problem. Therefore, it is useful to classify optimization problems with respect to certain categories. 

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It has been emphasized that the classes of objective function that have been outlined are by no means watertight however, provided they are not held to too rigidly they give a useful classification of optimal problems. In the following chapters we shall take up the study of different types of reactors and build up a sequence of useful cases which will show the particular features of the reactor system and lead to a discussion of general problems. It should be remembered however that the methods used for one particular reaction-reactor system and objective function are not necessarily suitable for another. At every turn there is scope for all the ingenuity and acumen the optimizer can muster. 

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