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 n.

  • "MSK_DOMAIN_RZERO": the zero domain, consisting of the origin 0nRn. Valid aliases include “ZERO” and “RZERO”.

  • "MSK_DOMAIN_RPLUS": the nonnegative orthant domain R0n. A valid alias is “RPLUS”.

  • "MSK_DOMAIN_RMINUS": the nonpositive orthant domain R0n. A valid alias is “RMINUS”.

  • "MSK_DOMAIN_R": the free domain, consisting of the whole Rn. A valid alias is “R”.

Membership in a linear domain is equivalent to imposing the corresponding set of n linear constraints, for instance Fx+g0n is equivalent to Fx+g=0 and so on. The free domain imposes no restriction.

13.6.2 Quadratic cone domains

The quadratic domains are determined by the dimension n.

Qn={xRn : x1x22++xn2}.

Valid aliases include “QUAD” and “QUADRATIC_CONE”.

Qrn={xRn : 2x1x2x32++xn2, x1,x20}.

Valid aliases include “RQUAD” and “RQUADRATIC_CONE”.

13.6.3 Exponential cone domains

Kexp={(x1,x2,x3)R3 : x1x2exp(x3/x2), x1,x20}.

Valid aliases include “PEXP” and “PRIMAL_EXP_CONE”.

Kexp={(x1,x2,x3)R3 : x1x3exp(x2/x31), x10,x30}.

Valid aliases include “DEXP” and “DUAL_EXP_CONE”.

13.6.4 Power cone domains

A power cone domain is determined by the dimension n and a sequence of 1nl<n positive real numbers (weights) α1,,αnl.

Pn(α1,,αnl)={xRn : i=1nlxiβixnl+12++xn2, x1,,xnl0}.

where βi are the weights normalized to add up to 1, ie. βi=αi/(jαj) for i=1,,nl. The name nl 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”.

(Pn(α1,,αnl))={xRn : i=1nl(xiβi)βixnl+12++xn2, x1,,xnl0}.

where βi are the weights normalized to add up to 1, ie. βi=αi/(jαj) for i=1,,nl. The name nl 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 nl=2 and weights (α,1α) for some 0<α<1 specified as cone parameter.

13.6.5 Geometric mean cone domains

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

GMn={xRn : (i=1n1xi)1/(n1)|xn|, x1,,xn10}.

It is a special case of the primal power cone domain with nl=n1 and weights α=(1,,1).

A valid alias is “PRIMAL_GEO_MEAN_CONE”.

(GMn)={xRn : (n1)(i=1n1xi)1/(n1)|xn|, x1,,xn10}.

It is a special case of the dual power cone domain with nl=n1 and weights α=(1,,1).

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 n, which must be of the form n=d(d+1)/2. Then the domain is defined as

S+d,vec={(x1,,xd(d+1)/2)Rn : sMat(x)S+d},

where

sMat(x)=[x1x2/2xd/2x2/2xd+1x2d1/2xd/2x2d1/2xd(d+1)/2],

or equivalently

S+d,vec={sVec(X) : XS+d},

where

sVec(X)=(X11,2X21,,2Xd1,X22,2X32,,Xdd).

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”.