The primal and dual pair of conic optimization problems over a self-dual cone are defined as

- is a closed, convex cone in a euclidean space .
- is a linear operator, and is its adjoint.
- , and .

**The space :**, where- are symmetric matrices (possibly of various sizes).
- are vectors (again, possibly of various sizes).
- is a vector.

**The cone :**, where- means that the symmetric matrix is positive semidefinite.
- means that the vector is in a quadratic cone (also known as the second-order cone, Lorentz cone or ice cream cone) of appropriate size. That is, if , then .
- means that the vector is componentwise nonnegative.

**The inner product :**For- For matrices and , .
- For vectors and , .

and

Thus the feasible set is a product of semidefinite, quadratic and nonnegative orthant cones, intersected with an affine subspace. It is possible that one or more of the three parts of the problem is absent, i.e., any of , , or the length of may be zero.

The *rotated quadratic cone* is defined as