In the previous post we discussed how we can project onto the epigraph of the squared norm. However, when in an optimisation problem we encounter constraints of the form

That is, quadratic constraints (of the form ), these can be converted to second-order conic constraints.

Note that in the constraints may or may not be a decision variable (it may as well be a constant).

The following identity does the trick:

Then,

where is the second-order cone, also known as Lorentz cone or ice cream cone.

Define the linear operator

Then, the quadratic constraints becomes

Having for some symmetric positive semidefinite matrix is no different than what we just did since .

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