second-order cone programming#Solvers and scripting (programming) languages

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

$\backslash mathcal\{C\}\_\{n+1\}=\backslash left\backslash \{\; \backslash begin\{bmatrix\}\; x\; \backslash \backslash \; t\; \backslash end\{bmatrix\}\; \backslash Bigg|\; x\; \backslash in\; \backslash mathbb\{R\}^n,$

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 $\backslash mathbb\{R\}^3$ is $\backslash left\backslash \{(x,y,z)\; \backslash Big|\; \backslash sqrt\{x^2\; +\; y^2\}\; \backslash leq\; z\; \backslash right\backslash \}$.

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

$\backslash lVert\; A\_i\; x\; +\; b\_i\; \backslash rVert\_2\; \backslash leq\; c\_i^T\; x\; +\; d\_i\; \backslash Leftrightarrow$

\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

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

$\backslash lVert\; A\_i\; x\; +\; b\_i\; \backslash rVert\_2\; \backslash leq\; c\_i^T\; x\; +\; d\_i\; \backslash Leftrightarrow$

\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.

File:Hierarchy compact convex.png

When $A\_i\; =\; 0$ for $i\; =\; 1,\backslash dots,m$, the SOCP reduces to a linear program. When $c\_i\; =\; 0$ for $i\; =\; 1,\backslash dots,m$, 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. 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. The converse, however, is not valid: there are positive semidefinite cones that do not admit any second-order cone representation.{{Cite journal|last=Fawzi|first=Hamza|date=2019|title=On representing the positive semidefinite cone using the second-order cone|journal=Mathematical Programming|language=en|volume=175|issue=1–2|pages=109–118|doi=10.1007/s10107-018-1233-0|issn=0025-5610|arxiv=1610.04901|s2cid=119324071}}

Any closed convex semialgebraic set in the plane can be written as a feasible region of a SOCP,.{{cite arXiv|last=Scheiderer|first=Claus|date=2020-04-08|title=Second-order cone representation for convex subsets of the plane|class=math.OC|eprint=2004.04196}} However, it is known that there exist convex semialgebraic sets of higher dimension that are not representable by SDPs; that is, there exist convex semialgebraic sets that can not be written as the feasible region of a SDP (nor, a fortiori, as the feasible region of a SOCP).{{Cite journal|last=Scheiderer|first=Claus|date=2018|title=Spectrahedral Shadows|journal=SIAM Journal on Applied Algebra and Geometry|language=en|volume=2|issue=1|pages=26–44|doi=10.1137/17M1118981|issn=2470-6566|doi-access=free}}

Consider a convex quadratic constraint of the form

:$x^T\; A\; x\; +\; b^T\; x\; +\; c\; \backslash leq\; 0.$

This is equivalent to the SOCP constraint

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

Consider a stochastic linear program in inequality form

:minimize $\backslash \; c^T\; x\; \backslash$

:subject to

::$\backslash mathbb\{P\}(a\_i^Tx\; \backslash leq\; b\_i)\; \backslash geq\; p,\; \backslash quad\; i\; =\; 1,\backslash dots,m$

where the parameters $a\_i\; \backslash$ are independent Gaussian random vectors with mean $\backslash bar\{a\}\_i$ and covariance $\backslash Sigma\_i\; \backslash$ and $p\backslash geq0.5$. This problem can be expressed as the SOCP

:minimize $\backslash \; c^T\; x\; \backslash$

:subject to

:: $\backslash bar\{a\}\_i^T\; x\; +\; \backslash Phi^\{-1\}(p)\; \backslash lVert\; \backslash Sigma\_i^\{1/2\}\; x\; \backslash rVert\_2\; \backslash leq\; b\_i\; ,\; \backslash quad\; i\; =\; 1,\backslash dots,m$

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

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.{{Cite journal |last=Alzalg |first=Baha M. |date=2012-10-01 |title=Stochastic second-order cone programming: Applications models |url=https://www.sciencedirect.com/science/article/pii/S0307904X11008547 |journal=Applied Mathematical Modelling |language=en |volume=36 |issue=10 |pages=5122–5134 |doi=10.1016/j.apm.2011.12.053 |issn=0307-904X|url-access=subscription }}

Other modeling examples are available at the MOSEK modeling cookbook.{{Cite web |title=MOSEK Modeling Cookbook - Conic Quadratic Optimization |url=https://docs.mosek.com/modeling-cookbook/cqo.html}}

• Power cones are generalizations of quadratic cones to powers other than 2.{{Cite web |title=MOSEK Modeling Cookbook - the Power Cones |url=https://docs.mosek.com/modeling-cookbook/powo.html}}

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Category:Optimization algorithms and methods

Category:Convex optimization