Limitations of SDP Semidefinite programming seems to be a very powerful tool to approach NP-hard optimisation problems, for example in discrete optimisation and there are some very interesting results (like the max cut approximation). What is know on the limitations of such approaches?
 A: First, finding the relaxation is the first challenge. It is not always straightforward to find an SDP relaxation. 
Second, Although you can solve the SDP relaxation exactly, establishing a guarantee that the solution to SDP relaxation has any relation to the solution of your NP-hard problem is extremely challenging.  This has been done beautifully for maxcut, but for many problems, there is no guarantees. For instance, to the best of my knowledge, SDP relaxation for sensor selection does not have a guarantee for sensor selection problem:
https://web.stanford.edu/~boyd/papers/pdf/sensor_selection.pdf. Another example is the positive principal component analysis problem where you can easily write the SDP, but so far there is no guarantee (see section 9.13 in https://cims.nyu.edu/~bandeira/TenLecturesFortyTwoProblems.pdf).
Third, for combinatorial problems such as maxcut and sensor selection, SDP relaxation produces a continuous solution and finding a good rounding procedure with provable guarantees is a difficult task. Again, this has been done very beautifully for maxcut.
Lastly, solving SDP exactly in many large-scale problems is extremely inefficient and you end up solving it approximately.
