Second-order cone programming

A second-order cone program (SOCP) is a convex optimization problem of the form

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

subject to

${\displaystyle \lVert A_{i}x+b_{i}\rVert _{2}\leq c_{i}^{T}x+d_{i},\quad i=1,\dots ,m}$

${\displaystyle Fx=g\ }$

where the problem parameters are ${\displaystyle f\in \mathbb {R} ^{n},\ A_{i}\in \mathbb {R} ^{{n_{i}}\times n},\ b_{i}\in \mathbb {R} ^{n_{i}},\ c_{i}\in \mathbb {R} ^{n},\ d_{i}\in \mathbb {R} ,\ F\in \mathbb {R} ^{p\times n}}$, and ${\displaystyle g\in \mathbb {R} ^{p}}$. ${\displaystyle x\in \mathbb {R} ^{n}}$ is the optimization variable. ${\displaystyle \lVert x\rVert _{2}}$ is the Euclidean norm and ${\displaystyle ^{T}}$ indicates transpose. The "second-order cone" in SOCP arises from the constraints, which are equivalent to requiring the affine function ${\displaystyle (Ax+b,c^{T}x+d)}$ to lie in the second-order cone in ${\displaystyle \mathbb {R} ^{n_{i}+1}}$.

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.