10.5 Supported domains

This section lists the domains supported by MOSEK.

10.5.1 Linear domains

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

• : the zero domain, consisting of the origin $0^n\in {\mathbb{R}}^n$.

• : the nonnegative orthant domain ${\mathbb{R}}_{\ge 0}^n$.

• : the nonpositive orthant domain ${\mathbb{R}}_{\le 0}^n$.

• : the free domain, consisting of the whole ${\mathbb{R}}^n$.

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.

10.5.2 Quadratic cone domains

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

• : the quadratic cone domain is the subset of ${\mathbb{R}}^n$ defined as

$${\mathcal{Q}}^n=\{x\in {\mathbb{R}}^n:x_1\ge \sqrt{x_2^2+\cdots +x_n^2}\}.$$

• : the rotated quadratic cone domain is the subset of ${\mathbb{R}}^n$ defined as

$${\mathcal{Q}}_r^n=\{x\in {\mathbb{R}}^n:2x_1{x}_2\ge x_3^2+\cdots +x_n^2,x_1,x_2\ge 0\}.$$

10.5.3 Exponential cone domains

• : the primal exponential cone domain is the subset of ${\mathbb{R}}^3$ defined as

$$K_{\exp }=\{(x_1,x_2,x_3)\in {\mathbb{R}}^3:x_1\ge x_2\exp (x_3/x_2),x_1,x_2\ge 0\}.$$

• : the dual exponential cone domain is the subset of ${\mathbb{R}}^3$ defined as

$$K_{\exp }^{\ast }=\{(x_1,x_2,x_3)\in {\mathbb{R}}^3:x_1\ge -x_3\exp (x_2/x_3-1),x_1\ge 0,x_3\le 0\}.$$

10.5.4 Power cone domains

A power cone domain is determined by the dimension $n$ and a sequence of $1\le n_l<n$ positive real numbers (weights) ${\alpha }_1,\dots ,{\alpha }_{n_l}$.

• : the primal power cone domain is the subset of ${\mathbb{R}}^n$ defined as

$${\mathcal{P}}_n^{({\alpha }_1,\dots ,{\alpha }_{n_l})}=\{x\in {\mathbb{R}}^n:\prod _{i=1}^{n_l}{x}_i^{{\beta }_i}\ge \sqrt{x_{n_l+1}^2+\cdots +x_n^2},x_1,\dots ,x_{n_l}\ge 0\}.$$

where ${\beta }_i$ are the weights normalized to add up to $1$, ie. ${\beta }_i={\alpha }_i/(\sum _j{\alpha }_j)$ for $i=1,\dots ,n_l$. 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 ${\mathbb{R}}^n$ defined as

$${\left({\mathcal{P}}_n^{({\alpha }_1,\dots ,{\alpha }_{n_l})}\right)}^{\ast }=\{x\in {\mathbb{R}}^n:\prod _{i=1}^{n_l}{\left(\frac{x_i}{{\beta }_i}\right)}^{{\beta }_i}\ge \sqrt{x_{n_l+1}^2+\cdots +x_n^2},x_1,\dots ,x_{n_l}\ge 0\}.$$

where ${\beta }_i$ are the weights normalized to add up to $1$, ie. ${\beta }_i={\alpha }_i/(\sum _j{\alpha }_j)$ for $i=1,\dots ,n_l$. 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.

10.5.5 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 ${\mathbb{R}}^n$ defined as

$${\mathcal{GM}}^n=\{x\in {\mathbb{R}}^n:{\left(\prod _{i=1}^{n-1}{x}_i\right)}^{1/(n-1)}\ge |x_n|,x_1,\dots ,x_{n-1}\ge 0\}.$$

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

• : the dual geometric mean cone domain is the subset of ${\mathbb{R}}^n$ defined as

$$({\mathcal{GM}}^n{)}^{\ast }=\{x\in {\mathbb{R}}^n:(n-1){\left(\prod _{i=1}^{n-1}{x}_i\right)}^{1/(n-1)}\ge |x_n|,x_1,\dots ,x_{n-1}\ge 0\}.$$

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

10.5.6 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

$${\mathcal{S}}_{+}^{d,\mathrm{vec}}=\{(x_1,\dots ,x_{d(d+1)/2})\in {\mathbb{R}}^n:\mathrm{sMat}(x)\in {\mathcal{S}}_{+}^d\},$$

where

$$\begin{array}{r}\mathrm{sMat}(x)=\left[\begin{array}{cccc}{x}_1& x_2/\sqrt{2}& \cdots & x_d/\sqrt{2}\\ x_2/\sqrt{2}& x_{d+1}& \cdots & x_{2d-1}/\sqrt{2}\\ \cdots & \cdots & \cdots & \cdots \\ x_d/\sqrt{2}& x_{2d-1}/\sqrt{2}& \cdots & x_{d(d+1)/2}\end{array}\right],\end{array}$$

or equivalently

$${\mathcal{S}}_{+}^{d,\mathrm{vec}}=\{\mathrm{sVec}(X):X\in {\mathcal{S}}_{+}^d\},$$

where

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

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.