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For questions about computationally solving a problem subject to some constraints on the solution. For optimizations, apply the more specific [constrained-optimization] tag instead.
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Semi-Definite relaxation of non-linear constraint?
square both side ?
(1-k)^2*trace(A*X)^2 == k^2*trace(B*X)
(1-k)^2*trace(A*X)^2 - k^2*trace(B*X) == 0
move from SDP to QCQP.
Or move to unconstrained problem with last term penalized with quadratic …