cone: Cone Constructors
In ROI: R Optimization Infrastructure

Constructor functions for the different cone types. Currently ROI supports eight different types of cones.

Zero cone

\mathcal{K}_{\mathrm{zero}} = \{0\}
Nonnegative (linear) cone

\mathcal{K}_{\mathrm{lin}} = \{x|x ≥q 0 \}
Second-order cone

\mathcal{K}_{\mathrm{soc}} = ≤ft\{(t, x) \ | \ ||x||_2 ≤q t, x \in R^n, t \in R \right\}
Positive semidefinite cone

\mathcal{K}_{\mathrm{psd}} = ≤ft\{ X \ | \ min(eig(X)) ≥q 0, \ X = X^T, \ X \in R^{n \times n} \right\}
Exponential cone

\mathcal{K}_{\mathrm{expp}} = ≤ft\{(x,y,z) \ | \ y e^{\frac{x}{y}} ≤q z, \ y > 0 \right\}
Dual exponential cone

\mathcal{K}_{\mathrm{expd}} = ≤ft\{(u,v,w) \ | \ -u e^{\frac{v}{u}} ≤q e w, u < 0 \right\}
Power cone

\mathcal{K}_{\mathrm{powp}} = ≤ft\{(x,y,z) \ | \ x^α * y^{(1-α)} ≥q |z|, \ x ≥q 0, \ y ≥q 0 \right\}
Dual power cone

\mathcal{K}_{\mathrm{powd}} = ≤ft\{ (u,v,w) \ | \ ≤ft(\frac{u}{α}\right)^α * ≤ft(\frac{v}{(1-α)}\right)^{(1-α)} ≥q |w|, \ u ≥q 0, \ v ≥q 0 \right\}

K_zero(size)

K_lin(size)

K_soc(sizes)

K_psd(sizes)

K_expp(size)

K_expd(size)

K_powp(alpha)

K_powd(alpha)

`size`	a integer giving the size of the cone, if the dimension of the cones is fixed (i.e. `zero`, `lin`, `expp`, `expd`) the number of cones is sufficient to define the dimension of the product cone.
`sizes`	a integer giving the sizes of the cones, if the dimension of the cones is not fixed (i.e. `soc`, `psd`) we have to define the sizes of each single cone.
`alpha`	a numeric vector giving the `alphas` for the (dual) power cone.