14.9 Supported domains

This section lists the domains supported by MOSEK. See Sec. 7 (Optimization Tutorials) for how to apply domains to specify conic constraints and disjunctive constraints (DJCs).

14.9.1 Affine domains

• Domain.equalsTo (==): the fixed domain consisting of a single point,

• Domain.lessThan (<=): the upper-bounded domain specified by an upper bound in each dimension,

• Domain.greaterThan (>=): the lower-bounded domain specified by a lower bound in each dimension,

• Domain.inRange: the ranged domain specified by an interval in each dimension,

• Domain.unbounded: the unbounded domain $\mathbb{R}$.

Membership in an affine domain imposes linear constraints in the model. The unbounded domain imposes no restriction.

14.9.2 Quadratic cone domains

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

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

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

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

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

14.9.3 Exponential cone domains

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

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

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

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

14.9.4 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) ${\alpha }_1,\dots ,{\alpha }_{n_l}$.

• Domain.inPPowerCone: 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.

• Domain.inDPowerCone: 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.

14.9.5 Geometric mean cone domains

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

• Domain.inPGeoMeanCone: 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 $\alpha =(1,\dots ,1)$.

• Domain.inDGeoMeanCone: 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 $\alpha =(1,\dots ,1)$.

14.9.6 Positive semidefinite cone domain

• Domain.inPSDCone is the domain ${\mathcal{S}}_{+}^d$ of symmetric positive-semidefinite variables of a given dimension $d$. It can only be applied to objects of shape $(d,d)$.