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I'll like to know if the constraint x_1 * x_2 >= x_3 * x_4 can be represented as an SOCP constraint.

Note that setting matrix A = [ x_1 x_3 \\ x_4 x_2] the constraint is equivalent to det(A) >= 0, so it is an SDP constraint.

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Unfortunately the set described by your constraint is not convex, which prevents both SDP and SOCP formulations. To see this, you can for example intersect it with hyperplanes x1=1 and x2=2: the resulting set x3*x4 <= 1 is clearly not convex, which implies that the original set was not either. The "det(A)>=0" argument invoked above does not work because matrix A is not symmetric.

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