# Supported Cone Types

Clarabel natively supports optimization problems with conic constraints defined on the following cones:

Cone TypeConstructorDefinition
Zero coneZeroConeT(n)$\{ 0 \}^{n}$
Nonnegative OrthantNonnegativeConeT(n)$\{ x \in \mathbb{R}^{n} : x_i \ge 0, \forall i=1,\dots,\mathrm{n} \}$
Second-Order ConeSecondOrderConeT(n)$\{ (t,x) \in \mathbb{R}^{n} : ||x||_2 \leq t \}$
Exponential ConeExponentialConeT()$\{(x, y, z) : y > 0,~~ ye^{x/y} ≤ z \}$
Power ConePowerConeT(a)$\{(x, y, z) : x^a y^{(1-a)} \geq |z|,~ (x,y) \geq 0 \}$ with $a \in (0,1)$
Generalized Power ConeGenPowerConeT(a,n)$\{(x, y) \in \mathbb{R}^{len(a)} \times \mathbb{R}^{n} : \prod\limits_{a_i \in a} x_i^{a_i} \geq ||y||_2,~ x \ge 0 \}$ with $a_i \in (0,1)$ and $\sum a_i = 1$
Positive Semidefinite Cone (triangular part)PSDTriangleConeT(n)Upper triangular part of the positive semidefinite cone $\mathbb{S}^{n}_+$. The elements $x$ of this cone represent the columnwise stacking of the upper triangular part of a positive semidefinite matrix $X \in \mathbb{S}^{n}_+$, so that $x \in \mathbb{R}^d$ with $d = {n(n+1)}/{2}.$