Primal-dual potential reduction algorithm for symmetric programming problems with nonlinear objective functions

We consider a primal-dual potential reduction algorithm for nonlinear convex optimization problems over symmetric cones. The same complexity estimates as in the case of the linear objective function are obtained provided a certain nonlinear system of equations can be solved with a given accuracy. This generalizes the result of K. Kortanek, F. Potra and Y. Ye [7]. We further introduce a generalized Nesterov-Todd direction and show how it can be used to achieve a required accuracy (by solving the linearization of above mentioned nonlinear system) for a class of nonlinear convex functions satisfying scaling Lipschitz condition. This result is a far-reaching generalization of results of F. Potra, Y. Ye and J. Zhu [8], [9]. Finally, we show that a class of functions (which contains quantum entropy function) satisfies scaling Lipschitz condition.