Second-order cone programming

The standard or unit second-order cone of dimension ${\displaystyle n+1}$ is defined as

${\displaystyle {\mathcal {C}}_{n+1}=\left\{{\begin{bmatrix}x\\t\end{bmatrix}}{\Bigg |}x\in \mathbb {R} ^{n},t\in \mathbb {R} ,||x||_{2}\leq t\right\}}$.

The second-order cone is also known by quadratic cone, ice-cream cone, or Lorentz cone. The second-order cone in ${\displaystyle \mathbb {R} ^{3}}$ is ${\displaystyle \left\{(x,y,z){\Big |}{\sqrt {x^{2}+y^{2}}}\leq z\right\}}$.

The set of points satisfying a second-order cone constraint is the inverse image of the unit second-order cone under an affine mapping:

${\displaystyle \lVert A_{i}x+b_{i}\rVert _{2}\leq c_{i}^{T}x+d_{i}\Leftrightarrow {\begin{bmatrix}A_{i}\\c_{i}^{T}\end{bmatrix}}x+{\begin{bmatrix}b_{i}\\d_{i}\end{bmatrix}}\in {\mathcal {C}}_{n_{i}+1}}$

and hence is convex.

The second-order cone can be embedded in the cone of the positive semidefinite matrices since

${\displaystyle ||x||\leq t\Leftrightarrow {\begin{bmatrix}tI&x\\x^{T}&t\end{bmatrix}}\succcurlyeq 0,}$

i.e., a second-order cone constraint is equivalent to a linear matrix inequality (Here ${\displaystyle M\succcurlyeq 0}$ means ${\displaystyle M}$ is semidefinite matrix). Similarly, we also have,

${\displaystyle \lVert A_{i}x+b_{i}\rVert _{2}\leq c_{i}^{T}x+d_{i}\Leftrightarrow {\begin{bmatrix}(c_{i}^{T}x+d_{i})I&A_{i}x+b_{i}\\(A_{i}x+b_{i})^{T}&c_{i}^{T}x+d_{i}\end{bmatrix}}\succcurlyeq 0}$.

A hierarchy of convex optimization problems. (LP: linear program, QP: quadratic program, SOCP second-order cone program, SDP: semidefinite program, CP: cone program.)

When ${\displaystyle A_{i}=0}$ for ${\displaystyle i=1,\dots ,m}$, the SOCP reduces to a linear program. When ${\displaystyle c_{i}=0}$ for ${\displaystyle i=1,\dots ,m}$, the SOCP is equivalent to a convex quadratically constrained linear program.

Convex quadratically constrained quadratic programs can also be formulated as SOCPs by reformulating the objective function as a constraint.[4] Semidefinite programming subsumes SOCPs as the SOCP constraints can be written as linear matrix inequalities (LMI) and can be reformulated as an instance of semidefinite program.[4] The converse, however, is not valid: there are positive semidefinite cones that do not admit any second-order cone representation.[3] In fact, while any closed convex semialgebraic set in the plane can be written as a feasible region of a SOCP,[8] it is known that there exist convex semialgebraic sets that are not representable by SDPs, that is, there exist convex semialgebraic sets that can not be written as a feasible region of a SDP.[9]