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

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if it is C\ssurried, that the changes toward the optimum point are
sufficiently small, then only a small deviation is introduced.
The e1iminC\tion procedure takes place by partitioning the vari-
able set into "states" and "decisions." The state variables are
the selected variables which are to be eliminated by the T active
constraints. The decision variables are the remaining independent
variables which will be employed to seek the minimum value of the
objective function, The criteria for the partition include two
aspects:

1. All slack variables are taken as decisions unless no
other x-variable is available to be a state variable.
Since all ~t are identically equal to zero, when the
algorithm moves from the old point xO to the new one
xV in its search for the minimum there is a 50 percent
cmance that the ~t will become negative. This is a
violation of the problem constraints; and
2. Since the same basic reasoning applies to the x-vari-
a~les, the largest absolute valued variables are best
suited to be state variables.

In the computer code of the algorithm described in Appendix B,
the selection of states and decisions is undertaken in a much
more complex procedure to insure numerical stability.

After ,partitioning the x-vector into state and decision
variables, the variables can be relabeled s for states and d
for decisions. Equation 1 at the initial point XO can then be
written,

y = min y(s ,s ,.
1 2

. 'ST,d ,d ,
1 2

. ,dD) ..........,.'. (4)

in which D is the number of decision variables and equals
(N - T). Ln addition, the constraints listed in Eq. 3 can be
rewritten as:

!.(~'9) - :E. = Q .......................'................ (5)

The next step is to employ the chain rule of calculating the
total differential of y, In vector notation,

oy = (II y)
s

as + (lIdy) Cl9.

.............................. .

(6)

where the ~ymbo1 Cly is used to denote the total differential
rather than the standard notation of dy. This modification is
made so that the d can be reserved to denote the decision
variables.

The derivatives of the constraining functions can also be
written in vector form,

(IIS!.)a~ + (lId!.)Cl9. -a:E. = Q ...............,.............(7)

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