Consider the following maximization problem. <mtable columnalign="right left right left right

Cooper Doyle

Cooper Doyle

Answered question

2022-07-05

Consider the following maximization problem.
max x R n x T P x a i x i b i ,   i Z [ 1 , n ] ,
where x i is the i-th scalar component of vector x and P > 0 is positive definite. As the objective is a convex function, the maximizer should be where all constraints are active, i.e. at an extreme point of the domain of the form x i = a i or x i = b i . Is the global maximizer the one of such points such that its norm x T x is maximal?

Answer & Explanation

Karissa Macdonald

Karissa Macdonald

Beginner2022-07-06Added 12 answers

Maximizing x T x is not generally equivalent to maximizing x T P x . As a simple counterexample, take this choice of parameters for x R 2 .
P = [ 2 1 1 2 ]
1 x 1 2
1 x 2 2
We see that x T x is maximized at x = [ 2 2 ] , while x T P x is maximized at x = [ 2 1 ] , [ 1 2 ] .

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