15.10 Supported domains

This section lists the domains supported by MOSEK. See Sec. 6 (Optimization Tutorials) for how to apply domains to specify affine conic constraints (ACCs) and disjunctive constraints (DJCs).

15.10.1 Linear domains

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

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.

15.10.2 Quadratic cone domains

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

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

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

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

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

15.10.3 Exponential cone domains

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

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

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

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

15.10.4 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.

15.10.5 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 \(\alpha=(1,\ldots,1)\).

15.10.6 Vectorized semidefinite domain

  • MSK_appendsvecpsdconedomain : 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.