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   x2 + y2 – z2 < R2

Math

Share With

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

                          Min   f(x, y, z)

                 Subject to   x² + y² – z² < R².

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

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

        m(x² + y² - z² - R²) = 0,  m ³ 0.

 (a) Write the Kuhn-Tucker conditions for the following problem using Lagrange multipliers m₁ and m²:

Min   f(rÖR²+z² cosq, rÖR²+z² sinq, z)

Subject to                               r-1 £ 0

                                                   -r £ 0

                                                 Z, q ? R¹                                                    

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

Tucker conditions (*). Start by expressing m in terms of m₁ and m₂. There should be three cases to consider: when r = 0,

0 < r < 1, and r = 1.