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We introduce semi-perfect obstruction theory of a Deligne-Mumford stack $X$consisting of local perfect obstruction theories with weak comparisons onoverlaps. We show that semi-perfect obstruction theory shares similarproperties with perfect obstruction theory. We use semi-perfect obstructiontheory to construct virtual cycles of moduli of derived objects on Calabi-Yauthreefolds.  In the revision, we assume the existence of obstruction sheaf and put the(coarse moduli of the) intrinsic normal cone inside the obstruction sheaf. Thisavoids the issue of descent of bundle stacks of the local obstructioncomplexes.

Semi-Perfect Obstruction theory and DT Invariants of Derived Objects