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The goal is to solve the following maximization/minimization problem:

\begin{align*} &\max_y/\min_y y^TQy\\&\text{s.t.}(y-x)^\top V(y-x)\le\beta^2\end{align*} where the optimization variable $y\in\mathbb R^{n}$ and the constant vector $x\in \mathbb R^n$. Matrices $Q\in \mathbb R^{n\times n}$ and $V\in \mathbb R^{n\times n}$ are positive definite matrices. $\beta>0$ is a constant.

How to obtain the analytical expression of the optimal solution $y$ (non-trivally)?

It seems that the method in Maximizing a linear function over an ellipsoid does not work well.

Rodrigo de Azevedo
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OwnCandy
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    What do you mean by "the method [...] does not work well"? – Charlie Vanaret - the Uno guy Nov 18 '22 at 08:08
  • The question "Maximizing a linear function over an ellipsoid" is solved by Lagrangian approach in the offered website. The new question may not be solved by using Lagrangian approach. \ Website: https://math.stackexchange.com/questions/1832467/maximizing-a-linear-function-over-an-ellipsoid – OwnCandy Nov 18 '22 at 09:30
  • Have you tried using the same approach? Now your objective is a strictly convex quadratic, which you can differentiate the same way the linear objective was differentiated on the other page. – Charlie Vanaret - the Uno guy Nov 18 '22 at 09:32
  • Yes, if using the same approach, the Langrangian function can be rewritten as f(y)=y^TQy+\lambda[\beta^2-(y-x)^TV(y-x)]. Differentiating w.r.t. y, it follows that Qy+\lambdaV(x-y)=0, i.e. (Q-\lambda V)y=-\lambda Vx. The matrix (Q-\lambda V) may be a singular matrix. The value of y and \lambda may be not described by Langrangian approach. – OwnCandy Nov 18 '22 at 09:53
  • I think $\lambda$ is negative (dual of an upper bound), so it's fine :) – Charlie Vanaret - the Uno guy Nov 18 '22 at 11:50
  • Minimization is convex, but maximization is not. – Rodrigo de Azevedo Nov 19 '22 at 02:00
  • The (convex) minimization problem can be rewritten as a semidefinite program (SDP). Take a look at this – Rodrigo de Azevedo Nov 19 '22 at 02:05
  • Negative $\lambda$ implies that $Q-\lambda V$ is positive definite. Then we can know that the optimal solution statisfies $y^=-\lambda(Q-\lambda V)^{-1}Vx$ and nontrivally in the boundary of constraints, i.e. $(y^-x)^TV(y^*-x)=\beta^2$. It is equivalent to $x^T[\lambda (Q-\lambda V)^{-1}V+I_m]^TV[\lambda (Q-\lambda V)^{-1}V+I_m]x=\beta^2$. This is why I say that $\lambda$ may be not described by Langrangian approach. Thank you.【Reply to cvanaret】 – OwnCandy Nov 19 '22 at 07:43
  • Thank you for the reference about the reformulation of the convex minimization problem by SDP. 【Reply to Rodrigo de Azevedo】 – OwnCandy Nov 19 '22 at 07:47
  • @OwnCandy You can now answer your own question for the minimization case. – Rodrigo de Azevedo Nov 19 '22 at 11:19

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