WSP10809

• File Number: 8273.500
• Description: Colorado River Basin Salinity Control - Federal Agency Reports - EPA
• Basin: Colorado Mainstem
• Water Division: 5
• Date: 4/1/1978
• Title: Integrating Desalination and Agricultural Salinity Control Alternatives
• Water Supply Pro - Doc Type: Report/Study

Theoretical Development

l\)
00 Consider the problem in which the minimum value of the
~ objective function is sought subject to a set of constraining
00 functions. Writing this problem mathematically,

y = min Y(?f) ........................................... (1)

subject to,

!.(~) > 0 ..............................................(2)

where the notation y(x) denotes "as a function of the vector x."
The number of x variables is defined as N and the number of
constraints as K. The method of analysis depends largely upon
the structure of the constraints. When all the constraints are
inequalities and "loose" or "inactive" (strictly>) at the
initial feasible point xO, the problem is "unconstrained." In
the other case when either some of these functions are strict
equalities or when some of the inequalities are "tight" or
"active," the problem is referred to as "constrained." Although
both of the conditions may occur in the solution of a problem,
they require somewhat different approaches as the algorithm
progresses toward the optimum.

Elimination Procedure

The elimination nature of the technique is derived from the
fact that it is at least conceptually possible to employ only
the currently active constraints to eliminate some of the x's
from the problem, making it temporarily unconstrained. To begin,
define the number of active constraints as T and reorder the
constraint set so that the first T is the active constraint
with index t = 1, 2, ..., T. Further, introduce "slack"
variables to the active constraints so they take the form,

!.(~) - ! = Q ........................................... (3)

and become strict equalities, where! is the vector of slack
variables. The purpose of this transformation is that by
continual observation of the slack variables the distinction
between active and inactive functions can be determined. The
problem now contains N original variables plus T slack variables
which are related by T active constraints. If the constraints
are linear, T of the variables can be eliminated from the
objective function by the constraint expressions, making the
problem unconstrained. However, in the general situation,
the constraints are nonlinear, and it is not directly possible
to substitute for the dependent variables, but rather to first
linearize the functions by taking the first partial derivatives
with respect to the x variables. Even though the nonlinearity
may still exist due to the nature'of the terms in the constraints,

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