Unformatted text preview: Minimize z = —$1 + 2x2 — 3x4 subject to
$1 + 2333 — x4 2 1
.552 — 3:3 2 0
2x1 — 2x; — x4 2 —3
x3 + 2224 2 3
~3x1 + x4 2 —1
$1,562, $3,134 2 O 4. Let f and g be convex functions on a convex set S. Deﬁne a function h
by _
W?) = #06) + 89(90) for all x in S, Where 1' and s are positive real numbers. Prove that h is also
a convex function on 8', making it clear in your proof how you are making
use of the hypothesis that r and s are positive. 5. Consider the linear programming problem below (in standard form). (a)
Show that x = (2, 1, 3, 0, 2)T is a feasible solution to the problem. (b) Show
that d = (1, 3, 4, 1, 5)T is a direction of unboundedness for the problem. (c)
Use parts (a) and (b) to obtain a feasible solution 90’ = x+7d to the problem
such that the value of the objective function at 56’ equals —100. (d) Generalize
part (c) to show that the problem has no optimal solutions. Minimize z = 3:171 — 2123 subject to
2951 — $2 + x4 = 3
3561 — 21133 + $5 = 2
$1,$2,$37$4,$5 Z 0 ...
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 Spring '10
 Brown

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