Answers to assignment due on October 14
th
.
2.
Program 1D:
*
z
=
min
y b,
subject to
y A
≥
c .
To place this thing in the format of Program 12.1, we need to express it as a
maximization problem with nonnegative variables and equality constraints.
Let’s replace
y
by the difference
yp

yn
of two vectors of nonnegative variables, let’s
insert a vector of surplus variables to convert the inequalities into equations, and let’s
multiply the objective by
1.
This gives

*
z
=
max
(yn – yp) b
,
subject to
v
(yp – yn) A – w
=
c ,
yp
≥
0 ,
yn
≥
0 ,
w
≥
0 .

Taking the dual of the latter gives

*
z
=
min (c v) ,
subject to
yn:
 A v
≥
b ,
yp:
A v
≥ – b ,
w:
 v
≥
0 .
Substituting
x
=
 v
and switching the sense of the objective gives
*
z
=
max
c x ,
subject to
A x
=
b ,
x
≥
0 ,
which is Program 12.1
3.
Denote as
j
x
the number of units of food
j
that the consumer consumes.
(a)
The minimumcost diet that satisfies his nutritional needs is found by solving the
linear program:
Minimize
∑
=
n
1
j
j
j
x
c
,
subject to the constraints
i
y
∑
≥
=
n
1
j
i
j
ij
b
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 Fall '10
 ERICDENARDO
 Optimization, linear program, nonnegative variables, unbounded feasible regions, yp – yn

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