such methods can be considerable for problems with many variables and con-

straints, as is the case here. Additionally, if an "optimum" is found by such a method,

there is no guarantee that it is a global optimum.

Methods have also been developed for general nonlinear problems. Perhaps the

method most commonly referred to is Lagrange's method of undetermined multi-

pliers (Wilde and Beightler 1967). This method requires that the solution to a set of

nonlinear equations be found. The number of unknowns is equal to the number of

variables (pipe diameters and consumer control valve pressure losses) plus the

number of constraints. In our case this would result in very large systems of

nonlinear equations for all but the most trivial problems. The solution of large

systems of nonlinear equations can be a very difficult task, usually done by adapting

methods for the solution of linear equations. For this reason, Lagrange's method is

felt to be impractical for this problem.

The Generalized Reduced Gradient (GRG) method is a popular one used for

nonlinear constrained problems (Reklaitis et al. 1983). It is based on extending

methods used for linear problems to nonlinear problems. The basic concept of the

GRG method is to follow along the direction of a constraint subset while seeking

improvement in the objective function. By requiring some subset of the constraints

to be satisfied, the number of degrees of freedom of the problem can be effectively

reduced. When inequality constraints are present in the problem, as is the case here,

either an active set strategy must be adopted or slack variables must be introduced

for each constraint. Gill et al. (1981) indicate that GRG methods can encounter

difficulties when highly nonlinear constraints are involved, as is the case here.

Because methods developed for linear problems are being used for nonlinear

problems, it is necessary to iterate at each step to achieve a feasible design. The

Newton-Rapson method is used for this iteration and it becomes the main compu-

tation burden of the GRG method (Arora 1989). Other quasi-Newton methods have

been proposed, but they can cause other problems with this method (Arora 1989).

Vanderplaats (1984) indicates that convergence of the Newton-Rapson method may

be a problem when using the GRG method for highly nonlinear problems.

Another class of methods for general nonlinear constrained problems is the

penalty function methods (Rao 1984). These methods reduce the constrained

problem to an unconstrained problem that can then be solved using any of the

various methods suitable for such problems. With many variables, as we have here,

the multidimensional optimization problem that results can be quite time consum-

ing to solve. In addition, it's usually necessary to solve the problem repeatedly for

different values of the penalty parameter until some convergence criterion has been

met. A feasible starting point is required as is an initial value for the penalty

parameter and the multiplication factor that is used to adjust the penalty parameter.

In this problem the diameters of the pipe segments must take on discrete values

in the final solution, while other variables such as the consumer control valve

pressure losses are continuous. Such a problem, which has both discrete and

continuous variables, can be formulated as what's called a "mixed integer" problem

(Reklaitis et al. 1983). The methods described above can not be applied directly to

integer or mixed integer problems. They must be used in combination with another

technique, most notably the "branch-and-bound" approach, to find the solution for

the discrete variables. The branch-and-bound approach will be discussed later.

In search of a simpler and more efficient method than those described above, we

will proceed by starting with our optimal independent (unconstrained) design and

identifying methods to move from this design to one that satisfies all the constraints.

We will attempt to conduct this process of modifying the solution so that it satisfies

the constraints in a manner that will keep us as close as possible to the true globally

optimal design.

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