Question

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(1) Apply the Kuhn Tucker condition to solve a minimization problem C (x,-4)2 + (x2-4)2 2x,+3x, 26 3x, 2x, 2 12 X,X2 20 Minimize Subject to and

Answer 1

Guven trroblem ig, (4-4 -4 Minimige Susjet to 224 9 * 6 3%- 22, シ 12 and Solution:: cendiHons ane Kuhn - Tucken The m. ,2,. . n 0 e j=,2,. . m く0 ,J 1,2, Qく In Bun obelen, 2 32-6 = - 31 - 2 x2 -12 C From firat cemdition, :{-5-22 120 2x32 3 A2 0 (-4 A1 (2) (1) = 0 + 2A1- 3 A2 - 8 > 0 lso, - A2 0 (22-4 (s) 0 2

rum Secomd condiHon of kuhn-Tukan ai (2%, +8%2-6) = 0 -3-2-12) A 2 NoD wa hane seneral cundiHons Case: I 2 4-8= O (2) 222-8= 0 Put this valuu in 2 (4) 812- 6 = 14 9(4) -6 32 0 -3(4) -2(4-2 of Kuhn Tuken does not sahts fics Conditon i) Alo opHimal solutien in this caje. +9%2-18 (2x (-324 -0) x frem, -h 4x2-24 0 from 3-2 2 -12 = 0 - 42 5 a2 42 A28.4 (8.4) -6 = 0 Q4 6-. 7 - = 9.6 than Aol for A,, ^2. Put this

from, (9.6) 2A, - 3^2 - 8 = 0 2A-3A2 192 -8 = 0 (1 (a) frum (8.4) +3 A, -2^, -8 = 0 + 9A,- 2A2 - 8 = 0 16.8 x 2 2 (a) - 9 A2 9.8G + - 4 A2 16 5 A2 0 = from () 12 0 48 + 12 = 0 5 - 9.6 + 6 ( un tradichon to Condi tion () oF Kuhn Tuckon.) No opimal s0lution.

Case:- M frm 0, (a +2 A-8 = 0) x 3 forom , (22 +3 a1 -8 = 0) x 2 6 6 - 24 = 0 A22 A 16 + 6- 42 34-22 5 X 2 - 4 = 0 from &2 3%2 - 6 = 0 6 - 422- 3 + frum, - 4 O 2 20 + 4 3 3 21 72 =) 1 C this Value in Put B 14 4 0 10 -6 2 34 234 = 0 0

Also. Au terms of are 0. 2 Kuhn-Tukan Aalisfied condiHon find Noo from 2 A 3a 0. 144 39 312-144 2 A1 8 39. 16 8 28 Solution set for this froblum is x 2 39