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QC equivalence to SOCC |
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where the problem parameters are <math>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}</math>, and <math>g \in \mathbb{R}^p</math>. Here <math>x\in\mathbb{R}^n</math> is the optimization variable. When <math>A_i = 0</math> for <math>i = 1,\dots,m</math>, the SOCP reduces to a [[linear program]]. When <math>c_i = 0 </math> for <math>i = 1,\dots,m</math>, the SOCP is equivalent to a convex [[Quadratically constrained quadratic program]]. [[Semidefinite programming|Semidefinite programs]] subsumes SOCPs as the SOCP constraints can be written as [[Linear matrix inequality|Linear Matrix Inequalities]](LMI) and can be reformulated as an instance of semi definite program. SOCPs can be solved with great efficiency by [[interior point methods]].
==Example:
Consider a quadratic constraint on the form
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