# unstable optimizer, stable objective

I am trying to minimize a convex objective numerically using gradient descent. I select the starting point randomly. I repeat the experiment multiple times. The optimal objective value I get each time is quit the same, but the minimizer is very different. Is it natural? How should it be handled in experiments?

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@user25004, is your quadratic positive definite or merely positive semidefinite? Consider minimizing $f(x,y) = y^2$ over $(x,y)\in\mathbb R^2$. (Or even $f(x,y) = 10^{-6}x^2 + y^2$, what with the slow convergence of gradient descent...) –  Rahul Oct 10 '12 at 3:16
@Rahul Narain It is $AX+ AXBX$ where $A$ and $B$ are positive semidefinite. Minimization is over matrix $X$. –  user25004 Oct 10 '12 at 4:32