A new line-search step based on the Weierstrass $\wp$-function for minimizing a class of logarithmic barrier functions

Summary.

We define the notion of self-concordance of order two for the restriction $f$ of a logarithmic barrier function to a given line. Based on this notion we prove an inner approximation of the domain of $f$, as well as a lower bound of the distance from a point $t$ to the minimum of $f$. These results provide the theoretical tools to develop a simple and efficient search step for finding the minimum of the barrier function along a given line. The new bound on the size of the line-search step is better than the optimal bound known for the case of a self-concordant function (of order one). We conclude with some numerical examples that illustrate the promise of the new line-search step.