13.6 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).
13.6.1 Linear domains¶
Each linear domain is determined by the dimension
"MSK_DOMAIN_RZERO"
: the zero domain, consisting of the origin . Valid aliases include “ZERO” and “RZERO”."MSK_DOMAIN_RPLUS"
: the nonnegative orthant domain . A valid alias is “RPLUS”."MSK_DOMAIN_RMINUS"
: the nonpositive orthant domain . A valid alias is “RMINUS”."MSK_DOMAIN_R"
: the free domain, consisting of the whole . A valid alias is “R”.
Membership in a linear domain is equivalent to imposing the corresponding set of
13.6.2 Quadratic cone domains¶
The quadratic domains are determined by the dimension
"MSK_DOMAIN_QUADRATIC_CONE"
: the quadratic cone domain is the subset of defined as
Valid aliases include “QUAD” and “QUADRATIC_CONE”.
"MSK_DOMAIN_RQUADRATIC_CONE"
: the rotated quadratic cone domain is the subset of defined as
Valid aliases include “RQUAD” and “RQUADRATIC_CONE”.
13.6.3 Exponential cone domains¶
"MSK_DOMAIN_PRIMAL_EXP_CONE"
: the primal exponential cone domain is the subset of defined as
Valid aliases include “PEXP” and “PRIMAL_EXP_CONE”.
"MSK_DOMAIN_DUAL_EXP_CONE"
: the dual exponential cone domain is the subset of defined as
Valid aliases include “DEXP” and “DUAL_EXP_CONE”.
13.6.4 Power cone domains¶
A power cone domain is determined by the dimension
"MSK_DOMAIN_PRIMAL_POWER_CONE"
: the primal power cone domain is the subset of defined as
where
are the weights normalized to add up to , ie. for . The name reads as “n left”, the length of the product on the left-hand side of the definition. Valid aliases include “PPOW” and “PRIMAL_POWER_CONE”.
"MSK_DOMAIN_DUAL_POWER_CONE"
: the dual power cone domain is the subset of defined as
where
are the weights normalized to add up to , ie. for . The name reads as “n left”, the length of the product on the left-hand side of the definition. Valid aliases include “DPOW” and “DUAL_POWER_CONE”.
Remark: in MOSEK 9 power cones were available only in the special case with
and weights for some specified as cone parameter.
13.6.5 Geometric mean cone domains¶
A geometric mean cone domain is determined by the dimension
"MSK_DOMAIN_PRIMAL_GEO_MEAN_CONE"
: the primal geometric mean cone domain is the subset of defined as
It is a special case of the primal power cone domain with
and weights . A valid alias is “PRIMAL_GEO_MEAN_CONE”.
"MSK_DOMAIN_DUAL_GEO_MEAN_CONE"
: the dual geometric mean cone domain is the subset of defined as
It is a special case of the dual power cone domain with
and weights . A valid alias is “DUAL_GEO_MEAN_CONE”.
13.6.6 Vectorized semidefinite domain¶
"MSK_DOMAIN_SVEC_PSD_CONE"
: the vectorized PSD cone domain is determined by the dimension , which must be of the form . Then the domain is defined as
where
or equivalently
where
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.
A valid alias is “SVEC_PSD_CONE”.