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You don't need SDP for an easy task like this. $X$ are your points. Do a PCA on $Y Y', Y = X - \operatorname{mean}(X)$, reduce dimension to 2-dimensions (others dimensions are zeros). 3 points on the canonical plane, axis of ellipse are 1st and 2nd Principal Components.
The nonlinear conjugate gradient method will converge for a quadratic function in $N$ steps at most. You do not have a quadratic function, and have no guarantees on convergence. It can be helpful to reset your search direction every $N$ steps or to use a better method (i.e., Newton's method). A simple function that you could use to test would be a quadratic ...