Second-order cone programming: Difference between revisions

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== Second-order cone ==

The standard or unit '''second-order cone''' of dimension <math>n+1</math> is defined as

<math>\mathcal{C}_{n+1}=\left\{ \begin{bmatrix} x \\ t \end{bmatrix} \Bigg| x \in \mathbb{R}^n,

t\in \mathbb{R}, ||x||_2\leq t \right\}</math>.

The second-order cone is also known by '''quadratic cone'',' or '''ice-cream cone'',' or '''Lorentz cone'''. The second-order cone in <math>\mathbb{R}^3</math> is <math>\left\{(x,y,z) \Big| \sqrt{x^2 + y^2} \leq z \right\}</math>.

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