K_zero {ROI}

Cone Constructors

Description

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 \geq 0 \}

• Second-order cone

  \mathcal{K}_{\mathrm{soc}} = \left\{(t, x) \ | \ ||x||_2 \leq t, x \in R^n, t \in R \right\}

• Positive semidefinite cone

  \mathcal{K}_{\mathrm{psd}} = \left\{ X \ | \ min(eig(X)) \geq 0, \ X = X^T, \ X \in R^{n \times n} \right\}

• Exponential cone

  \mathcal{K}_{\mathrm{expp}} = \left\{(x,y,z) \ | \ y e^{\frac{x}{y}} \leq z, \ y > 0 \right\}

• Dual exponential cone

  \mathcal{K}_{\mathrm{expd}} = \left\{(u,v,w) \ | \ -u e^{\frac{v}{u}} \leq e w, u < 0 \right\}

• Power cone

  \mathcal{K}_{\mathrm{powp}} = \left\{(x,y,z) \ | \ x^\alpha * y^{(1-\alpha)} \geq |z|, \ x \geq 0, \ y \geq 0 \right\}

• Dual power cone

  \mathcal{K}_{\mathrm{powd}} = \left\{ (u,v,w) \ | \ \left(\frac{u}{\alpha}\right)^\alpha * \left(\frac{v}{(1-\alpha)}\right)^{(1-\alpha)} \geq |w|, \ u \geq 0, \ v \geq 0 \right\}

Usage

K_zero(size)

K_lin(size)

K_soc(sizes)

K_psd(sizes)

K_expp(size)

K_expd(size)

K_powp(alpha)

K_powd(alpha)

Arguments

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.

Examples

K_zero(3) ## 3 equality constraints
K_lin(3)  ## 3 constraints where the slack variable s lies in the linear cone