  The self-concordant-like property of a smooth convex function is a new
analytical structure that generalizes the self-concordant notion. While a wide
variety of important applications feature the self-concordant-like property,
this concept has heretofore remained unexploited in convex optimization. To
this end, we develop a variable metric framework of minimizing the sum of a
"simple" convex function and a self-concordant-like function. We introduce a
new analytic step-size selection procedure and prove that the basic gradient
algorithm has improved convergence guarantees as compared to "fast" algorithms
that rely on the Lipschitz gradient property. Our numerical tests with
real-data sets shows that the practice indeed follows the theory.
