A NEW LINE-SEARCH STEP BASED ON THE WEIERSTRASS S-C-FUNCTION FOR MINIMIZING A CLASS OF LOGARITHMIC BARRIER FUNCTIONS

We define the notion of self-concordance of order two for the restrict ion 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. The se results provide the theoretical tools to develop a simple and effic ient 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 be tter than the optimal bound known for the case of a self-concordant fu nction (of order one). We conclude with some numerical examples that i llustrate the promise of the new line-search step.