Unconstrained convex minimization in relative scale

In this paper we present a new approach to constructing schemes for unconstrained convex minimization, which compute approximate solutions with a certain relative accuracy. This approach is based on a special conic model of the unconstrained minimization problem. Using a structural model of the objective function we can employ the efficient smoothing technique. The fastest of our algorithms solves a linear programming problem with relative accuracy [delta] in at most e. sq.m(2 + lnm).(1 + 1 /[delta]) iterations of a gradient-type scheme, where m is the largest dimension of the problem and e is the Euler number.