Could you help me solve this problem please ? 1. Maximize x t </msup> y wit

Brooklynn Hubbard

Brooklynn Hubbard

Answered question

2022-05-09

Could you help me solve this problem please ?

1. Maximize x t y with constraint x t Q x 1 (where Q is definite positive)
What I tried : I tried using KKT but I don't know why I get y t Q 1 y as the maximum instead of y t Q 1 y (which I believe is the maximum). Also, since x t y is linear (convex and concave), I don't know how to conclude...
2. Conclude that ( x t y ) 2 ( x t Q x ) ( y t Q 1 y ) x , y (generalized CS)

Answer & Explanation

Jamal Hamilton

Jamal Hamilton

Beginner2022-05-10Added 11 answers

You might check your KKT calculations again; indeed, you should be getting y T Q 1 y as the maximum. Now since x T y is convex, it is good; it means that satisfying the KKT conditions is sufficient for finding your desired global maximum.

For part 2, note that by rearranging, equivalently you want to prove that
( x ~ T y ) 2 y T Q 1 y ,
where x ~ = x x T Q x . Now how can you use the solution from part 1 to deduce this?

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