Operations Research
NonLinear Programming
Problem 1. For the following NLP problems: identify all the KKT points, geometrically represent the set of
feasible solutions, determine the optimal solution and justify the answer.
(a) max z= 2x1+x2
s.t.: x2
1+x2
2⩽1
x1⩾0
(b) max z= (2x1−5)2+ (2x2−1)2
s.t.: x1+ 2x2⩽2
x1, x2⩾0
(c) min z= 2x1−x2
s.t.: −x2
1+x2⩽0
(x1−1)2+x2−5⩽0
−x2⩽0
(d) max z= 2x1−x2
s.t.: −x1+x2⩽0
(x1−1)2+x2−5⩽0
−x2⩽0
(e) min z= (x1−2)2+ (x2−2)2
s.t.: −x1+x2⩽0
(x1−1)2+x2−5⩽0
−x2⩽0
(f) min z=x2
1
−x2
2
s.t.: −(x1−2)2
−x2+ 4 ⩽0
(g) min z=x2
1+1
2x2
2
s.t.: x1−x2⩾0
x1+x2⩽0
x1⩽0
(h) min z=x2
1
−x2
2
s.t.: 2x1+ 3x2⩾1
2x1+ 3x2⩽0
(i) min z=x1+ 2x2
s.t.: 2x1+ 3x2⩾6
x1, x2⩾0