Second-order cone programming: Difference between revisions

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Line 16: ::<math>\lVert A x + b \rVert_2 \leq c^T x + d</math> These each define a subspace that is bounded by an inequality based on a [[Degree of a polynomial\|second-order ~~polynomisl~~polynomial]] function defined on the optimization variable <math>x</math>; this can be shown to define a [[convex cone]], hence the name "'''second-order cone'''".<ref>{{Cite journal \|~~last~~last1=Jibrin \|~~first~~first1=Shafiu \|last2=Swift \|first2=James W. \|date=2024 \|title=On Second-Order Cone Functions ~~\|url=https://onlinelibrary.wiley.com/doi/abs/10.1155/2024/7090058~~ \|journal=Journal of Optimization \|language=en \|volume=2024 \|issue=1 \|pages=7090058 \|doi=10.1155/2024/7090058 \|doi-access=free \|issn=2314-6486}}</ref> 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]]<ref>{{cite journal\|last1=Potra\|first1=lorian A.\|last2=Wright\|first2=Stephen J.\|date=1 December 2000\|title=Interior-point methods\|journal=Journal of Computational and Applied Mathematics\|volume=124\|issue=1–2\|pages=281–302\|doi=10.1016/S0377-0427(00)00433-7\|bibcode=2000JCoAM.124..281P\|doi-access=}}</ref> and in general, can be solved more efficiently than [[semidefinite programming]] (SDP) problems.<ref name="Fawzi" /> Some engineering applications of SOCP include filter design, antenna array weight design, truss design, and grasping force optimization in robotics.<ref name=":0">{{Cite journal\|last1=Lobo\|first1=Miguel Sousa\|last2=Vandenberghe\|first2=Lieven\|last3=Boyd\|first3=Stephen\|last4=Lebret\|first4=Hervé\|date=1998\|title=Applications of second-order cone programming\|journal=Linear Algebra and Its Applications\|language=en\|volume=284\|issue=1–3\|pages=193–228\|doi=10.1016/S0024-3795(98)10032-0\|doi-access=free}}</ref> 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.<ref>{{cite web \|title=Solving SOCP \|url=https://docs.mosek.com/slides/2017/shanghai/talk.pdf}}</ref><ref>{{cite web \|title=portfolio optimization \|url=https://nmfin.tech/wp-content/uploads/2020/06/new-technologies-in-portfolio-optimization.20200612.pdf}}</ref><ref>{{cite book \|last1=Li \|first1=Haksun \|title=Numerical Methods Using Java: For Data Science, Analysis, and Engineering \|date=16 January 2022 \|publisher=APress \|pages=Chapter 10 \|isbn=978-1484267967 }}</ref>