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¶
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 \(\real\).
Membership in an affine domain imposes linear constraints in the model. The unbounded domain imposes no restriction.
14.8.2 Quadratic cone domains¶
The quadratic domains are determined by the dimension \(n\).
Domain.inQCone: the quadratic cone domain is the subset of \(\real^n\) defined as
\[\Q^n = \left\{ x\in\real^n~:~ x_1 \geq \sqrt{x_2^2+\cdots+x_n^2} \right\}.\]
Domain.inRotatedQCone: the rotated quadratic cone domain is the subset of \(\real^n\) defined as
\[\Q_r^n = \left\{ x\in\real^n~:~ 2x_1x_2 \geq x_3^2+\cdots+x_n^2,\ x_1,x_2\geq 0 \right\}.\]
14.8.3 Exponential cone domains¶
Domain.inPExpCone: the primal exponential cone domain is the subset of \(\real^3\) defined as
\[\EXP = \left\{ (x_1,x_2,x_3)\in\real^3~:~ x_1 \geq x_2 \exp(x_3/x_2),\ x_1,x_2\geq 0\right\}.\]
Domain.inDExpCone: the dual exponential cone domain is the subset of \(\real^3\) defined as
\[\EXP^* = \left\{ (x_1,x_2,x_3)\in\real^3~:~ x_1 \geq -x_3 \exp(x_2/x_3-1),\ x_1\geq 0,x_3\leq 0\right\}.\]
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) \(\alpha_1,\ldots,\alpha_{n_l}\).
Domain.inPPowerCone: 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.
Domain.inDPowerCone: 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.
14.8.5 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 \(\real^n\) 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 \(\alpha=(1,\ldots,1)\).
Domain.inDGeoMeanCone: 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 \(\alpha=(1,\ldots,1)\).
14.8.6 Positive semidefinite cone domain¶
Domain.inPSDConeis the domain \(\PSD^d\) of symmetric positive-semidefinite variables of a given dimension \(d\). It can only be applied to objects of shape \((d,d)\).
