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# smid02 - Minimize z = —\$1 2x2 — 3x4 subject to \$1 2333...

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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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