# Trouble visualizing a second-order cone constraint

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.

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 ?

• Which book? Boyd & Vandenberghe's Convex Optimization? May 7, 2023 at 10:31
• Just use different dummy variables in the description of $C$ and it will become clear. May 7, 2023 at 10:34

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

• Thanks for your answer; I read the link. How can I get the intuition to what is going on in the transformation? Jan 16, 2020 at 1:57
• @Frank Well it's just arbitrary linear transformations. The important thing is that SOCP is a certain way of writing certain quadratic constraints and being able to recognize when to use it. Try not to overthink the general form but look at some concrete examples. Jan 16, 2020 at 17:35
• Thanks. Things like this drive me crazy, if you happen to run across a good chapter of a text that is specifically about cones, please let me know. Jan 17, 2020 at 1:56

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

Hope this may helpful to you.

I was also trying to imagine what second order constraint looks like and what does $$A$$, $$b$$, $$c$$, and $$d$$ control in the $$||Ax + b||_2 \leq c^Tx + d$$ constraint.

So, let us try to imagine this in 2D plane. We can rewrite this constraint as

$$||Ax-x_0||_2 \leq c^T(x - x_1)$$

Now, let us take example when $$A$$ is indentity and $$c$$ is $$(1, 0)$$ and $$x_1$$ is $$0$$, then the constraint means all those points, whose norm from $$x_0$$ is equal to distance from the directrix at origin!

Which is nothing but a parabola! now if $$c$$ was say $$(e,0)$$ then we can write that as $$e*(1,0)$$ and then we have ellipse (or hyperbola) with e as an eccentricity.

So $$c$$ controls the direction of the directrix, $$d$$ controls it's position and $$b$$ controls the position of the focii.

Now to understand what $$A$$ does. We can also write the constraint as

$$||Ax - x0||_2 \leq (c^TA^{*})Ax - x_1$$, $$A^*$$ is pseudo inverse

$$||x^\prime - x0||_2 \leq f^Tx^\prime - x_1$$ which is same thing just in a distorted (affine) coordinate system.