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enclinesbnnbk 2022-05-07 Answered
Let x R n and let P be a n × n positive definite symmetrix matrix. It is known that the maximum of
maximize x T P x subject to x T x 1
is λ max ( P ), the largest eigenvalue of P. Now consider the following problem
maximize x T P x subject to ( x a ) T ( x a ) 1
where a R n is known. What is the analytical solution?
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Answers (1)

Answered 2022-05-08 Author has 15 answers
This is (a variant of) the trust-region problem, and the solution can be computed rather easily, although not an analytical solution
I'm reparameterizing your problem slightly, you can easily change your model to be max x T Q x + c T x + b subject to x T x 1. At optimality, you will be at the border of feasibility, and the constraint gradient will be pointing in the same direction as the objective gradient (draw this geometrically!), i.e., you have 2 Q x + c = λ 2 x for some unknown λ. Solve for x to obtain x = ( 2 Q λ I ) 1 c. To find lambda, solve x T x = 1 which is an equation in λ (possibly tricky, numerical methods required, line-search, bisection etc)
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