# 14.8 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).

## 14.8.1 Affine domains¶

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

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

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

## 14.8.3 Exponential cone domains¶

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

## 14.8.4 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) $$\alpha_1,\ldots,\alpha_{n_l}$$.

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

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.

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

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.

## 14.8.5 Geometric mean cone domains¶

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

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

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

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

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

## 14.8.6 Positive semidefinite cone domain¶

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