Notes on duality in second order and p-order cone optimization

Recently, the so-called second order cone optimization problem has received much attention, because the problem has many applications and the problem can in theory be solved efficiently by interior-point methods. In this note we treat duality for second order cone optimization problems and in particular whether a nonzero duality gap can be obtained when casting a convex quadratically constrained optimization problem as a second order cone optimization problem. Furthermore, we also discuss the p-order cone optimization problem which is a natural generalization of the second order case. Specifically, we suggest a new self-concordant barrier for the p-order cone optimization problem.