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 or ice-cream cone or Lorentz cone. The standard 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,

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

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]

Quadratic constraint edit

Consider a convex quadratic constraint of the form

${\displaystyle x^{T}Ax+b^{T}x+c\leq 0.}$ 

This is equivalent to the SOCP constraint

${\displaystyle \lVert A^{1/2}x+{\frac {1}{2}}A^{-1/2}b\rVert \leq \left({\frac {1}{4}}b^{T}A^{-1}b-c\right)^{\frac {1}{2}}}$ 

Stochastic linear programming edit

Consider a stochastic linear program in inequality form

minimize ${\displaystyle \ c^{T}x\ }$ 

subject to

${\displaystyle \mathbb {P} (a_{i}^{T}x\leq b_{i})\geq p,\quad i=1,\dots ,m}$ 

where the parameters ${\displaystyle a_{i}\ }$  are independent Gaussian random vectors with mean ${\displaystyle {\bar {a}}_{i}}$  and covariance ${\displaystyle \Sigma _{i}\ }$  and ${\displaystyle p\geq 0.5}$ . This problem can be expressed as the SOCP

minimize ${\displaystyle \ c^{T}x\ }$ 

subject to

${\displaystyle {\bar {a}}_{i}^{T}x+\Phi ^{-1}(p)\lVert \Sigma _{i}^{1/2}x\rVert _{2}\leq b_{i},\quad i=1,\dots ,m}$ 

where ${\displaystyle \Phi ^{-1}(\cdot )\ }$  is the inverse normal cumulative distribution function.^[1]

Stochastic second-order cone programming edit

We refer to second-order cone programs as deterministic second-order cone programs since data defining them are deterministic. Stochastic second-order cone programs are a class of optimization problems that are defined to handle uncertainty in data defining deterministic second-order cone programs.^[10]

Other examples edit

Other modeling examples are available at the MOSEK modeling cookbook.^[11]

• Power cones are generalizations of quadratic cones to powers other than 2.^[14]