6 Conversions.
6.1 Minimization-maximization.
Consider a real-valued function :fC→R defined on some set C. Let :gC→R be given by
() ()
gx fx=−. If ˆ
xC∈ is a maximizer of f, then ˆ
x is a minimizer of g. To see this, observe
that
() ()
() ()
ˆˆ
gx fx fx gx=−≤−=
for all xC∈.
Exercise
Show that if ˆ
xC∈ is a minimizer of :fC→R, then ˆ
x is a maximizer of :gC→R, where g
is given by
() ()
gx fx=−.
6.2 Reversal of inequalities.
Inequalities of the form ax b≥ are converted to the standard form by multiplying both sides by
1−.
Exercise
First convert the standard dual problem to a standard primal problem. Next take the dual of the
converted problem. Do you recognize this problem?
6.3 Splitting equalities.
Equalities of the form Dx e= split into a pair Dx e≤ and Dx e≥. In the primal problem this
pair appears as
Dx e
Dx e
≤
−≤−
.
It is of course possible to use the system Dx e= and solve for some of the variables i
x in terms
of the others and then replace the i
x throughout the problem.
6.4 Unrestricted variables.
The restriction that all variables are non-negative is possible to circumvent by increasing the
number of variables. Suppose that i
x is allowed to be both positive and negative. Introduce
0
i
p≥ and 0
i
n≥. Replace i
x throughout the problem with iii
xpn=−. Observe how the
linearity is retained. In this way each converted unrestricted variable increases the dimension by
1.
