Standard Forms of SOCP

SOCP > SOC inequalities | Standard forms | Group sparsity | Applications

Second-order cone
Rotated second-order cone
Second-order cone inequalities

Second-order cone

The second-order cone in $mathbf{R}^{p+1}$ is defined as

$mathbf{K}_p := left{ (x,y) in mathbf{R}^{p+1} ~:~ |x|_2 le y right}.$

This set is convex, since it is the intersection of (an infinite number of) half-spaces:

$mathbf{K}_p = bigcap_{u ::: |u|_2 le 1} left{ (x,y) in mathbf{R}^{p+1}~:~ x^Tu le y right} .$

It is a cone, since it is invariant by scaling: if $x in mathbf{K}_p$ , so does alpha x for any alpha ge 0 .

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The second-order cone in $mathbf{R}^3$ . The set actually extends to infinity upwards. For some strange reason, this set is called an ‘‘ice-cream cone’’.

Example: Magnitude constraints on affine complex vectors.

Rotated second-order cone

The rotated second-order cone in $mathbf{R}^{p+2}$ is the set

$mathbf{K}_p := left{ (x,y,z) in mathbf{R}^{p+2} ~:~ x^Tx le yz, ;; y ge 0, ;; z ge 0 right}.$

Note that the rotated second-order cone in $mathbf{R}^{p+2}$ can be expressed as a linear transformation (actually, a rotation) of the (plain) second-order cone in $mathbf{R}^{p+2}$ , since

$|x|_2^2 le yz , ;; y ge 0, ;; z ge 0 Longleftrightarrow left| left( begin{array}{c} 2x y-z end{array} right) right|_2 le y+z.$

This is, $(x,y,z) in mathbf{K}_p^r$ if and only if $(w,y+z) in mathbf{K}_{p+1}$ , where w = (x,y-z) . This proves that rotated second-order cones are also convex.

Rotated second-order cone constraints are useful to describe quadratic convex inequalities. Precisely, if Q=Q^T succeq 0 , the constraint

c^Tx + x^TQx le t

is equivalent to the existence of w,y,z such that

$w^Tw le yz, ;; z = 1, ;; w = Q^{1/2}x, ;; y = t-c^Tx ,$

where $Q^{1/2}$ is the square-root of the PSD matrix . In the space of (x,w,y,) -variables, the above constraints represent the intersection of a rotated second-order cone with affine sets.

Second-order cone inequalities

A second-order cone (SOC) inequality on a vector $x in mathbf{R}^n$ states that a vector (y,t) that is some affine combination of belongs to a second-order cone.

This is a constraint of the form

|Ax+b |_2 le c^Tx + d,

where $A in mathbf{R}^{m times n}$ , $b in mathbf{R}^m$ , $c in mathbf{R}^n$ , and is a scalar.

Examples: