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Homework answers / question archive / (5) Let R > 0 and assume the following problem has a solution: Min f(x, y, z) Subject to x2 + y2 – z2 < R2

(5) Let R > 0 and assume the following problem has a solution:

Min f(x, y, z)

Subject to x^{2} + y^{2} – z^{2} __<__ R^{2}.

Note that the Kuhn-Tucker conditions (*) are:

Ñ f(x, y, z) + m [2x, 2y, -2z) = [0, 0, 0]

m(x^{2} + y^{2} - z^{2} - R^{2}) = 0, m ³ 0.

(a) Write the Kuhn-Tucker conditions for the following problem using Lagrange multipliers m_{1} and m^{2}:

Min f(rÖR^{2}+z^{2} cosq, rÖR^{2}+z^{2} sinq, z)

Subject to r-1 £ 0

-r £ 0

Z, q ? R^{1 }

(b) Show that the conditions you found in part (a) imply the Kuhn-

Tucker conditions (*). Start by expressing m in terms of m_{1} and m_{2}. There should be three cases to consider: when r = 0,

0 < r < 1, and r = 1.

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