On self-concordant barriers for generalized power cones

Abstract

In the study of interior-point methods for nonsymmetric conic optimization and their applications, Nesterov (Optim Methods Softw 27(4–5): 893–917, 2012) introduced a 4-self-concordant barrier for the power cone, also known as Koecher’s cone. In his PhD thesis, Chares (Cones and interior-point algorithms for structured convex optimization involving powers and exponentials, PhD thesis, Universite catholique de Louvain, 2009) found an improved 3-self-concordant barrier for the power cone. In addition, he introduced the generalized power cone, and conjectured a “nearly optimal” self-concordant barrier for it. In this paper, we prove Chares’ conjecture. As a byproduct of our analysis, we derive a self-concordant barrier for a nonnegative power cone.

Appendix: Derivative calculations

In this section, we work through the derivative calculations in the proofs of Theorems 1 and 2. All derivatives are taken at the point (x, z) and in direction $(\varDelta x, \varDelta z)$. The first two derivatives of $\xi$ are easy to compute as $\xi ' = e_1 \xi$ and $\xi '' = -e_2 \xi$. Before computing $\xi '''$, we compute $e_1'$ and $e_2'$, which are useful quantities in computing $\xi '''$ as well as other derivatives later. Noting that $\delta _i' = -\delta _i^2$ (where $\delta _i$ is as in the proofs of Theorems 1 and 2), it is straightforward to show

$$\begin{aligned} e_1' = -s_2 = -e_1^2 - e_2 \end{aligned}$$

and

$$\begin{aligned} e_2' = s_2' - 2 s_1 s_1' = -2 s_3 + 2s_1 s_2 = -e_1 e_2 - e_3. \end{aligned}$$

We then compute $\xi '''$ by

$$\begin{aligned} \xi ''' = (-e_2 \xi )' = -e_2' \xi -e_2 \xi ' = (e_1 e_2 + e_3) \xi - e_2 (e_1 \xi ) = e_3 \xi . \end{aligned}$$

Now we compute the derivatives of the function $f = \xi - \frac{\Vert z\Vert _2^2}{\xi }$ in the proof of Theorem 1:

$$\begin{aligned} f'&= \xi ' + \frac{\xi }{\xi ^2} \Vert z \Vert _2^2 - \frac{1}{\xi } \left( 2z \cdot \varDelta z \right) = \xi ' + \frac{1}{\xi } ( e_1 \Vert z \Vert _2^2 - 2 z \cdot \varDelta z) ,\\ f''&= \left( \xi ' + \frac{1}{\xi } ( e_1 \Vert z \Vert _2^2 - 2 z \cdot \varDelta z) \right) ' \\&= \xi '' - \frac{\xi '}{\xi ^2} (e_1 \Vert z \Vert _2^2 - 2 z \cdot \varDelta z) + \frac{1}{\xi } \left( (-e_1^2 - e_2) \Vert z \Vert _2^2 + 2 e_1 z \cdot \varDelta z - 2 \Vert \varDelta z \Vert _2^2 \right) \\&= \xi '' - \frac{1}{\xi } \left( 2 e_1^2 \Vert z \Vert _2^2 - 4 e_1 z \cdot \varDelta z + e_2 \Vert z \Vert _2^2 + 2 \Vert \varDelta z \Vert _2^2 \right) \\&= \xi '' - \frac{e_2}{\xi } \Vert z \Vert _2^2 - \frac{2}{\xi } \Vert e_1 z - \varDelta z \Vert _2^2 ,\\ f'''&= \left( \xi '' - \frac{e_2}{\xi } \Vert z \Vert _2^2 - \frac{2}{\xi } \Vert e_1 z - \varDelta z \Vert _2^2 \right) ' \\&= \xi ''' + \frac{\xi '}{\xi ^2} e_2 \Vert z \Vert _2^2 - \frac{1}{\xi } \left( (-e_1 e_2 - e_3) \Vert z\Vert _2^2 + 2 e_2 z \cdot \varDelta z \right) + \frac{2 \xi '}{\xi ^2} \Vert e_1 z - \varDelta z \Vert _2^2 \\&- \frac{2}{\xi } \left( 2 e_1 (-e_1^2 - e_2) \Vert z\Vert _2^2 + 2 e_1^2 z \cdot \varDelta z - 2 \left( (-e_1^2 - e_2) z \cdot \varDelta z + e_1 \Vert \varDelta z \Vert _2^2 \right) \right) \\&= \xi ''' + \frac{e_3}{\xi } \Vert z \Vert _2^2 + \frac{2}{\xi } \left( 3 e_1 e_2 \Vert z\Vert _2^2 - 3 e_2 z \cdot \varDelta z + e_1 \Vert e_1 z - \varDelta z \Vert _2^2 \right. \\&\quad \left. + 2 e_1^3 \Vert z\Vert _2^2 - 4 e_1^2 z \cdot \varDelta z + 2 e_1 \Vert \varDelta z \Vert _2^2 \right) \\&= \xi ''' + \frac{e_3}{\xi } \Vert z \Vert _2^2 + \frac{2}{\xi } \left( 3 e_1 e_2 \Vert z\Vert _2^2 - 3 e_2 z \cdot \varDelta z + 3 e_1 \Vert e_1 z - \varDelta z \Vert _2^2 \right) \\&= \xi ''' + \frac{e_3}{\xi } \Vert z \Vert _2^2 + \frac{6}{\xi } \left( e_1 \Vert e_1 z - \varDelta z \Vert _2^2 + e_2 z \cdot (e_1 z - \varDelta z) \right) . \end{aligned}$$

The derivatives of $z - \frac{z^2}{\xi }$ in the proof of Theorem 2 are computed in a very similar manner and are omitted.