13.8 Supported domains

This section lists the domains supported by MOSEK.

13.8.1 Linear domains

Each linear domain is determined by the dimension \(n\).

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

• "nonnegative" : the nonnegative orthant domain \(\real_{\geq 0}^n\).

• "nonpositive" : the nonpositive orthant domain \(\real_{\leq 0}^n\).

• "unbounded" : 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.

13.8.2 Quadratic cone domains

The quadratic domains are determined by the dimension \(n\).

• "quad" : the quadratic cone domain is the subset of \(\real^n\) defined as

  \[\Q^n = \left\{ x\in\real^n~:~ x_1 \geq \sqrt{x_2^2+\cdots+x_n^2} \right\}.\]

• "rquad" : the rotated quadratic cone domain is the subset of \(\real^n\) defined as

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

13.8.3 Exponential cone domains

• "exp" : the primal exponential cone domain is the subset of \(\real^3\) defined as

  \[\EXP = \left\{ (x_1,x_2,x_3)\in\real^3~:~ x_1 \geq x_2 \exp(x_3/x_2),\ x_1,x_2\geq 0\right\}.\]

• "dexp" : the dual exponential cone domain is the subset of \(\real^3\) defined as

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

13.8.4 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" : 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.

• "dpow" : 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.

13.8.5 Geometric mean cone domains

A geometric mean cone domain is determined by the dimension \(n\).

• "geomean" : 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 \(\alpha=(1,\ldots,1)\).

• "dgeomean" : 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 \(\alpha=(1,\ldots,1)\).