Does any one know the exact formula(s) for convex objective functions which their solution is SVD decomposition of matrix X?
I suppose you refer to the optimisation problem of finding a linear combination of p vectors of length n that have maximum variance (or sum of squares when directly linking it with SVD) ..and then adding up an uncorrelated constraints for the next linear combination (having maximum variance but being uncorrelated with the first one). This usually leads to to eigen-decomposition of the Xt(X) (with your notation).
You may also have a look at the bilinear optimisation, equation (5) in the following paper, expressed using a tensor formalism.
Leibovici, D.G. (2010) "Spatio-temporal Multiway Decomposition using Principal Tensor Analysis on k-modes: the R package PTAk." Journal of Statistical Software, 34(10), 1-34. I think the paper is uploaded on my ResearchGate.
didier
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