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 0nRn.

  • "nonnegative" : the nonnegative orthant domain R0n.

  • "nonpositive" : the nonpositive orthant domain R0n.

  • "unbounded" : the free domain, consisting of the whole Rn.

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.

13.8.2 Quadratic cone domains

The quadratic domains are determined by the dimension n.

  • "quad" : the quadratic cone domain is the subset of Rn defined as

    Qn={xRn : x1x22++xn2}.
  • "rquad" : the rotated quadratic cone domain is the subset of Rn defined as

    Qrn={xRn : 2x1x2x32++xn2, x1,x20}.

13.8.3 Exponential cone domains

  • "exp" : the primal exponential cone domain is the subset of R3 defined as

    Kexp={(x1,x2,x3)R3 : x1x2exp(x3/x2), x1,x20}.
  • "dexp" : the dual exponential cone domain is the subset of R3 defined as

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

13.8.4 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.

  • "pow" : the primal power cone domain is the subset of Rn defined as

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.

  • "dpow" : the dual power cone domain is the subset of Rn defined as

(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.

  • 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.

13.8.5 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 Rn defined as

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).

  • "dgeomean" : the dual geometric mean cone domain is the subset of Rn defined as

(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).