Setting aside many of the nuances of this problem, it looks like you want to determine if a HALT oracle can be used to compute the limit of a sequence of SDP optimal objective values. Is this the case?

If so, is there anything in your particular problem that makes this fundamentally different from using a HALT oracle to compute a limit of a plain-vanilla sequence of real numbers? Your sequence would have some nice properties (i.e. a limit is known to exist; the sequence is monotonic) that a HALT oracle could exploit.

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