Trouble visualizing a second-order cone constraint

Question

I have this problem from Boyd's optimization textbook:

A standard form for the SOCP model is $$\text{minimize } f^Tx \\\text{subject to: }‖A_ix+b_i‖_2 ≤ c_i^Tx + d_i, i= 1,\dots,m$$ where we see that the variables $(A_ix+b_i,c^T_ix+d_i)$ should belong to a second-order cone of appropriate size.

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I don't see why the variables are in a second-order cone. According to the text, a second-order cone is the set

$$C = \left\{ (x, t) \in \Bbb R^{n+1} : \| x \|_2 \leq t \right\}$$

is $t$ the same as $c_i^Tx + d_i$? Can someone tip me off to how this is a cone? It seems like a set of all hyperplanes less than or equal to a line for each i ?

Answer 1

The cone $C$ can be visualized as in https://docs.mosek.com/modeling-cookbook/cqo.html#quadratic-cones. In 3 dimensions it really is like an ice-cream cone.

And then instead of the pair $(t,x)$ you have the pair $(c^Tx+d,Ax+b)$, and the inequality they satisfy remains the same, so it is just some affine transformation of the cone $C$ (squeezing, stretching, shifting, rotating). See also https://docs.mosek.com/modeling-cookbook/cqo.html#second-order-cones

Answer 2

I tried to code it: Code and record it: VideoLink

Hope this may helpful to you.