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I have a polynomial in x, that also depends on {y1,y2,y3,y4,y5}. It always has 10 (not necessarily distinct) real roots. It's top coefficient is (1*x^10) I aim to find the y's that maximize the smallest root. I know of the Minimize command, but I haven't worked with roots before.

Actually, the polynomial is the determinant of a 10*10 symmetric matrix of the form (M + Ix) + (N1 y1 + N2 y2 .. + N5 y5) . So it can be restated as maximizing its smallest eigenvalue.

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    $\begingroup$ It seems more profitable to treat this as an symmetric eigenvalue maximization problem than a generic root maximization problem. There seems to be a bunch of specialized methods for this route, but I have no experience with them. $\endgroup$ Commented Oct 5, 2018 at 10:56
  • $\begingroup$ Is this a "matrix pencil" problem? It might be worth investigating related techniques. $\endgroup$
    – mikado
    Commented Oct 5, 2018 at 13:30
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    $\begingroup$ It's hard to say without an example, but you could try NMaximize[Min[Eigenvalues[matrix]], {x, y1, y2, y3, y4, y5}] $\endgroup$
    – Carl Woll
    Commented Oct 5, 2018 at 15:45
  • $\begingroup$ Perhaps, you could provide a simplified problem, perhaps a 4x4 matrix with two parameters, for readers to solve. By the way, remember that the value of the minimum eigenvalue is only piecewise an analytical function of the parameters. $\endgroup$
    – bbgodfrey
    Commented Oct 5, 2018 at 17:47
  • $\begingroup$ The NMaximize arpoach is what I ended up going for. It seems to work. (unless it was a local maximum, but that's irrelevant. I've figured out the behavior I was interested in) $\endgroup$ Commented Oct 9, 2018 at 12:13

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