Unconstrained Convex Minimization in Relative Scale

Abstract:     
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 δ in at most e · √m(2 + lnm) · (1 + 1 δ ) iterations of a gradient-type scheme, where m is the largest dimension of the problem and e is the Euler number.

Number of Pages in PDF File: 20

Keywords: nonlinear optimization, convex optimization, complexity bounds, relative accuracy, fully polynomial approximation schemes, gradient methods, optimal methods

working papers series