| # 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 is defined as This set is convex, since it is the intersection of (an infinite number of) half-spaces: It is a cone, since it is invariant by scaling: if , so does for any .
| The second-order cone in . 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 is the set Note that the rotated second-order cone in can be expressed as a linear transformation (actually, a rotation) of the (plain) second-order cone in , since This is, if and only if , where . This proves that rotated second-order cones are also convex. Rotated second-order cone constraints are useful to describe quadratic convex inequalities. Precisely, if , the constraint is equivalent to the existence of such that where is the square-root of the PSD matrix . In the space of -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 states that a vector that is some affine combination of belongs to a second-order cone. This is a constraint of the form where , , , and is a scalar. Examples: