A second-order cone program (SOCP) is a convex optimization problem of the form
minimize f T x {\displaystyle \ f^{T}x\ }
subject to
‖ A i x + b i ‖ 2 ≤ c i T x + d i , i = 1 , … , m {\displaystyle \lVert A_{i}x+b_{i}\rVert _{2}\leq c_{i}^{T}x+d_{i},\quad i=1,\dots ,m}
F x = g {\displaystyle Fx=g\ }
where the problem parameters are f ∈ R n , A i ∈ R n i × n , b i ∈ R n i , c i ∈ R n , d i ∈ R , F ∈ R p × n {\displaystyle f\in \mathbb {R} ^{n},\ A_{i}\in \mathbb {R} ^{{n_{i}}\times n},\ b_{i}\in \mathbb {R} ^{n_{i}},\ c_{i}\in \mathbb {R} ^{n},\ d_{i}\in \mathbb {R} ,\ F\in \mathbb {R} ^{p\times n}} , and g ∈ R p {\displaystyle g\in \mathbb {R} ^{p}} . x ∈ R n {\displaystyle x\in \mathbb {R} ^{n}} is the optimization variable.
‖ x ‖ 2 {\displaystyle \lVert x\rVert _{2}} is the Euclidean norm and T {\displaystyle ^{T}} indicates transpose. The name "second-order cone programming" comes from the nature of the individual constraints, which are each of the form:
‖ A x + b ‖ 2 ≤ c T x + d {\displaystyle \lVert Ax+b\rVert _{2}\leq c^{T}x+d}
These each define a subspace that is bounded by an inequality based on a second-order polynomial function defined on the optimization variable x {\displaystyle x} ; this can be shown to define a convex cone, hence the name "second-order cone". By the definition of convex cones, their intersection can also be shown to be a convex cone, although not necessarily one that can be defined by a single second-order inequality. See below for a more detailed treatment. SOCPs can be solved by interior point methods and in general, can be solved more efficiently than semidefinite programming (SDP) problems. Some engineering applications of SOCP include filter design, antenna array weight design, truss design, and grasping force optimization in robotics. Applications in quantitative finance include portfolio optimization; some market impact constraints, because they are not linear, cannot be solved by quadratic programming but can be formulated as SOCP problems.
Second-order cones The standard or unit second-order cone of dimension n + 1 {\displaystyle n+1} is defined as
C n + 1 = { [ x t ] | x ∈ R n , t ∈ R , ‖ x ‖ 2 ≤ t } {\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 the names quadratic cone, ice-cream cone, or Lorentz cone. For example, the standard second-order cone in R 3 {\displaystyle \mathbb {R} ^{3}} is
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