Problems 1172.71 Consider the following problem:
Maximizef (x)=(x− 1 )^2subject to
− 2 ≤x≤ 4Determine whether the constraint qualification and Kuhn–Tucker conditions are satisfied
at the optimum point.2.72 Consider the following problem:
Minimizef=(x 1 − 1 )^2 +(x 2 − 1 )^2subject to
2 x 2 −( 1 −x 1 )^3 ≤ 0
x 1 ≥ 0
x 2 ≥ 0Determine whether the constraint qualification and the Kuhn–Tucker conditions are sat-
isfied at the optimum point.2.73 Verify whether the following problem is convex:
Minimizef (X)= − 4 x 1 +x^21 − 2 x 1 x 2 + 2 x 22subject to
2 x 1 +x 2 ≤ 6
x 1 − 4 x 2 ≤ 0
x 1 ≥ 0 , x 2 ≥ 02.74 Check the convexity of the following problems.
(a) Minimizef (X)= 2 x 1 + 3 x 2 −x 13 − 2 x^22subject to
x 1 + 3 x 2 ≤ 6
5 x 1 + 2 x 2 ≤ 10
x 1 ≥ 0 , x 2 ≥ 0(b) Minimizef (X)= 9 x 12 − 18 x 1 x 2 + 13 x 1 − 4subject to
x^21 +x^22 + 2 x 1 ≥ 162.75 Identify the optimum point among the given design vectors,X 1 ,X 2 , andX 3 , by applying
the Kuhn–Tlucker conditions to the following problem:
Minimizef (X)= 100 (x 2 −x 12 )^2 +( 1 −x 1 )^2