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If the constrained optimization problem is a quadratically constrained quadratic program of the form \begin{align}\min&\quad x^HQx-a(x+x^H)+b|z^Hx|^2\\\text{s.t.}&\quad\|x\|^2\le1\end{align} where

  1. $x$ is the unknown complex vector
  2. $Q$ is a positive semidefinite matrix
  3. $a$ is a scalar
  4. $b$ is a scalar
  5. $z$ is a known complex vector

What implication does a changing $Q$ have on the optimal value of $x$, i.e., what kind of change in $Q$ will result in a different value of $x$? Because even as $Q$ changes I am not able to see any change in $f(x)$.

Thank you.

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  • $\begingroup$ What is $H$? Are all entries of the $z$ vector nonnegative? If yes $x=0$ might always be the optimal solution with $f(x) =0$. $\endgroup$ – Michael Feldmeier May 22 at 7:47
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    $\begingroup$ @MichaelFeldmeier: Since $x$ is complex, I assume $x^H$ is the Hermitian transpose of $x$. Also, since $z$ is complex, I doubt it is nonnegative. $\endgroup$ – prubin May 22 at 19:58
  • $\begingroup$ Hi @Kali, welcome to OR.SE! I have two questions: 1) is something missing a the $(x+x^H)$ part? More specifically, how do we add $x$ and $x^H$ and multiply the result with scalar $a$ to get a real number? And 2) is the $Q$ matrix composed of real or of complex numbers? If the latter, is $Q$ Hermitian? $\endgroup$ – dhasson May 23 at 23:00
  • $\begingroup$ PS: Could you clarify on what kind of result you wish to find? For example, will any kind of result help, or do you need some sort of sensitivity analysis? examples of sensitivity analysis: if ratio of variation of the optimal $x$'s components when a specific coefficient $q_{ij}$ of matrix $Q$ changes, or ratio of variation in $x$ as the norm of $Q$ changes. $\endgroup$ – dhasson May 24 at 4:50

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