Turning a problem into a dual version is mostly a mechanical thing—you can immediately write it down in the case of LPs, for example[0], and many other convex cases [1].

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[0] https://en.m.wikipedia.org/wiki/Dual_linear_program

[1] http://www.seas.ucla.edu/~vandenbe/236C/lectures/conj.pdf slide 23 on down.