CO250 INTRODUCTION TO OPTIMIZATION  SOLUTIONS 11
Solution 1.
(a) The feasible region is the convex region in the middle of the following diagram, with extreme
points
(0
,
2)
,
(1
,
3)
,
(2
,
3)
and
(2
,
0)
. The cone of tight constraints at one of these points consists of the
set of vectors pointing from the extreme point into the shaded region which touches the feasible region
at that point (these regions extend to inﬁnity in the appropriate directions).
(b) Solving
(

2
,
1) =
y
1
(

1
,

1) +
y
2
(

1
,
1)
for
y
1
and
y
2
, we ﬁnd
y
1
=
1
2
,
y
2
=
3
2
.
(c) The dual (D) is
min
(

2
,
2
,
2
,
3)
y
subject to

y
1

y
2
+
y
3
=

2

y
1
+
y
2
+
y
4
=
1
y
≥
0
So we may set
y
1
and
y
2
to the solution of (b), and
y
i
= 0
for
y
i
corresponding to the nontight
constraints at this extreme point. In this way, we arrive at a solution of (D) to be
y
= (
1
2
,
3
2
,
0
,
0)
.
One easily checks that this really is feasible for (D), and and the objective function has value
2
at this
point. This is equal to the objective function of (P) at the extreme point in question,
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 Spring '10
 GUENIN
 Vector Space, Convex set, Convex function, extreme point, Geodesic convexity

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