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** Second order cone programming (SOCP)** problems are a type of

*convex optimization*problems. The general form of the problem is

\[\begin{array}{ll}

\mbox{minimize} & \ f^T x \\

\mbox{subject to} & \lVert A_i x + b_i \rVert_2 \leq c_i^T x + d_i,\quad i = 1,\dots,m \\

& Fx = g

\end{array} \]

where \(f \in \mathbb{R}^n\), \(A_i \in \mathbb{R}^{{n_i}\times n}\), \(b_i \in \mathbb{R}^{n_i}\), \(c_i \in \mathbb{R}^n\), \(d_i \in \mathbb{R}\), \(F \in \mathbb{R}^{p\times n}\), and \(g \in \mathbb{R}^p\). The inequalities, \(\lVert A_i x + b_i \rVert_2 \leq c_i^T x + d_i\) are the

*second order cone constraints*.

**Special cases**

- When \(A_i = 0\) for \(i = 1,\dots,m\), the SOCP reduces to a linear programming problem.
- When \(c_i = 0\) for \(i = 1,\dots,m\), the SOCP is equivalent to a convex quadratically constrained quadratic programming problem.

Note that semidefinite programming subsumes second order cone programming since the SOCP constraints can be written as linear matrix inequalities.

## Online and Software Resources

- Second Order Cone Programming Solvers on the NEOS Server
- SOCP example GAMS models
- Existing Multi-Facility Location Problem: emfl.gms
- Linear Phase Lowpass Filter Design Problem: fdesign.gms
- Mean-Variance Model with Variable Upper and Lower Bounds: pmeanvar.gms

## References

- Alizadeh, F. and D. Goldfarb. 2003. Second-order cone programming.
*Mathematical Programming***95**, 3 - 51. - Boyd, S. and Vandenberghe, L. 2004.
*Convex Optimization*. Cambridge University Press. - Optimization Online Linear, Cone, and Semidefinite Programming area