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Consider the problem min (x + y3 + z³) s.t. x² +y² + z² = 1 Given a condition that a point (x, y, z) is a KKT point of the problem if and only if x² + y² + z² = 1 2hx, y² = 21y, and z? = 21z. and there exists 1 ER satisfying x (1)Suppose (x, y, z) is a KKT point and A the corresponding multiplier from the condition above. Show that x + y + z + 0and ^ 2(x+y+z) (2)Find the optimal solution(s) to the problem. (hint: you may need to consider separately the three cases when one, two, or none of the variables (x, y, z) are zero.)

Consider the problem min (x + y3 + z³) s.t. x² +y² + z² = 1 Given a condition that a point (x, y, z) is a KKT point of the problem if and only if x² + y² + z² = 1 2hx, y² = 21y, and z? = 21z. and there exists 1 ER satisfying x (1)Suppose (x, y, z) is a KKT point and A the corresponding multiplier from the condition above. Show that x + y + z + 0and ^ 2(x+y+z) (2)Find the optimal solution(s) to the problem. (hint: you may need to consider separately the three cases when one, two, or none of the variables (x, y, z) are zero.)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Consider the problem min (x + y³ + z³) s.t. x² + y² + z? = 1 Given a condition that a point (x, y, z) is a KKT point of the problem if and only if x2 +y² + z² = 1 and there exists 1 E R satisfying x = 2hx, y² = 21y, and z? = 2hz. (1)Suppose (x, y, z) is a KKT point and A the corresponding multiplier from the condition above. Show that x + y + z + 0and 1 2(x+y+z) (2)Find the optimal solution(s) to the problem. (hint: you may need to consider separately the three cases when one, two, or none of the variables (x, y, z) are zero.)
Transcribed Image Text:Consider the problem min (x + y³ + z³) s.t. x² + y² + z? = 1 Given a condition that a point (x, y, z) is a KKT point of the problem if and only if x2 +y² + z² = 1 and there exists 1 E R satisfying x = 2hx, y² = 21y, and z? = 2hz. (1)Suppose (x, y, z) is a KKT point and A the corresponding multiplier from the condition above. Show that x + y + z + 0and 1 2(x+y+z) (2)Find the optimal solution(s) to the problem. (hint: you may need to consider separately the three cases when one, two, or none of the variables (x, y, z) are zero.)
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