# 11.5 Supported domains¶

This section lists the domains supported by MOSEK.

## 11.5.1 Linear domains¶

Each linear domain is determined by the dimension $$n$$.

• : the zero domain, consisting of the origin $$0^n \in\real^n$$.

• : the nonnegative orthant domain $$\real_{\geq 0}^n$$.

• : the nonpositive orthant domain $$\real_{\leq 0}^n$$.

• : the free domain, consisting of the whole $$\real^n$$.

Membership in a linear domain is equivalent to imposing the corresponding set of $$n$$ linear constraints, for instance $$Fx+g\in 0^n$$ is equivalent to $$Fx+g=0$$ and so on. The free domain imposes no restriction.

The quadratic domains are determined by the dimension $$n$$.

• : the quadratic cone domain is the subset of $$\real^n$$ defined as

$\Q^n = \left\{ x\in\real^n~:~ x_1 \geq \sqrt{x_2^2+\cdots+x_n^2} \right\}.$
• : the rotated quadratic cone domain is the subset of $$\real^n$$ defined as

$\Q_r^n = \left\{ x\in\real^n~:~ 2x_1x_2 \geq x_3^2+\cdots+x_n^2,\ x_1,x_2\geq 0 \right\}.$

## 11.5.3 Exponential cone domains¶

• : the primal exponential cone domain is the subset of $$\real^3$$ defined as

$\EXP = \left\{ (x_1,x_2,x_3)\in\real^3~:~ x_1 \geq x_2 \exp(x_3/x_2),\ x_1,x_2\geq 0\right\}.$
• : the dual exponential cone domain is the subset of $$\real^3$$ defined as

$\EXP^* = \left\{ (x_1,x_2,x_3)\in\real^3~:~ x_1 \leq -x_3 \exp(x_2/x_3-1),\ x_1\geq 0,x_3\leq 0\right\}.$

## 11.5.4 Power cone domains¶

A power cone domain is determined by the dimension $$n$$ and a sequence of $$1\leq n_l<n$$ positive real numbers (weights) $$\alpha_1,\ldots,\alpha_{n_l}$$.

• : the primal power cone domain is the subset of $$\real^n$$ defined as

$\POW_n^{(\alpha_1,\ldots,\alpha_{n_l})} = \left\{ x\in\real^n~:~ \prod_{i=1}^{n_l}x_i^{\beta_i} \geq \sqrt{x_{n_l+1}^2+\cdots+x_n^2},\ x_1,\ldots,x_{n_l}\geq 0 \right\}.$

where $$\beta_i$$ are the weights normalized to add up to $$1$$, ie. $$\beta_i=\alpha_i/(\sum_j \alpha_j)$$ for $$i=1,\ldots,n_l$$. The name $$n_l$$ reads as “n left”, the length of the product on the left-hand side of the definition.

• : the dual power cone domain is the subset of $$\real^n$$ defined as

$\left(\POW_n^{(\alpha_1,\ldots,\alpha_{n_l})}\right)^* = \left\{ x\in\real^n~:~ \prod_{i=1}^{n_l}\left(\frac{x_i}{\beta_i}\right)^{\beta_i} \geq \sqrt{x_{n_l+1}^2+\cdots+x_n^2},\ x_1,\ldots,x_{n_l}\geq 0 \right\}.$

where $$\beta_i$$ are the weights normalized to add up to $$1$$, ie. $$\beta_i=\alpha_i/(\sum_j \alpha_j)$$ for $$i=1,\ldots,n_l$$. The name $$n_l$$ reads as “n left”, the length of the product on the left-hand side of the definition.

• Remark: in MOSEK 9 power cones were available only in the special case with $$n_l=2$$ and weights $$(\alpha,1-\alpha)$$ for some $$0<\alpha<1$$ specified as cone parameter.

## 11.5.5 Geometric mean cone domains¶

A geometric mean cone domain is determined by the dimension $$n$$.

• : the primal geometric mean cone domain is the subset of $$\real^n$$ defined as

$\GM^n = \left\{ x\in\real^n~:~ \left(\prod_{i=1}^{n-1}x_i\right)^{1/(n-1)} \geq |x_n|,\ x_1,\ldots,x_{n-1}\geq 0 \right\}.$

It is a special case of the primal power cone domain with $$n_l=n-1$$ and weights $$\alpha=(1,\ldots,1)$$.

• : the dual geometric mean cone domain is the subset of $$\real^n$$ defined as

$(\GM^n)^* = \left\{ x\in\real^n~:~ (n-1)\left(\prod_{i=1}^{n-1}x_i\right)^{1/(n-1)} \geq |x_n|,\ x_1,\ldots,x_{n-1}\geq 0 \right\}.$

It is a special case of the dual power cone domain with $$n_l=n-1$$ and weights $$\alpha=(1,\ldots,1)$$.

## 11.5.6 Vectorized semidefinite domain¶

• : the vectorized PSD cone domain is determined by the dimension $$n$$, which must be of the form $$n=d(d+1)/2$$. Then the domain is defined as

$\PSD^{d,\mathrm{vec}} = \left\{(x_1,\ldots,x_{d(d+1)/2})\in \real^n~:~ \mathrm{sMat}(x)\in\PSD^d\right\},$

where

$\begin{split}\mathrm{sMat}(x) = \left[\begin{array}{cccc}x_1 & x_2/\sqrt{2} & \cdots & x_{d}/\sqrt{2} \\ x_2/\sqrt{2} & x_{d+1} & \cdots & x_{2d-1}/\sqrt{2} \\ \cdots & \cdots & \cdots & \cdots \\ x_{d}/\sqrt{2} & x_{2d-1}/\sqrt{2} & \cdots & x_{d(d+1)/2}\end{array}\right],\end{split}$

or equivalently

$\PSD^{d,\mathrm{vec}} = \left\{\mathrm{sVec}(X)~:~X\in\PSD^d\right\},$

where

$\mathrm{sVec}(X) = (X_{11},\sqrt{2}X_{21},\ldots,\sqrt{2}X_{d1},X_{22},\sqrt{2}X_{32},\ldots,X_{dd}).$

In other words, the domain consists of vectorizations of the lower-triangular part of a positive semidefinite matrix, with the non-diagonal elements additionally rescaled.