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An algorithm for LC1 optimization

In this paper an algorithm for LC1 unconstrained optimization problems, which uses the second order Dini upper directional derivative is considered. The purpose of the paper is to establish general algorithm hypotheses under which convergence occurs to optimal points. A convergence proof is given, as well as an estimate of the rate of convergence. | Yugoslav Journal of Operations Research 15 (2005), Number 2, 301-306 AN ALGORITHM FOR LC1 OPTIMIZATION* Nada I. ĐURANOVIĆ-MILIČIĆ Department of Mathematics, Faculty of Technology and Metallurgy University of Belgrade, Belgrade, Yugoslavia nmilicic@elab.tmf.bg.ac.yu Received: January 2003 / Accepted: March 2005 Abstract. In this paper an algorithm for LC1 unconstrained optimization problems, which uses the second order Dini upper directional derivative is considered. The purpose of the paper is to establish general algorithm hypotheses under which convergence occurs to optimal points. A convergence proof is given, as well as an estimate of the rate of convergence. Keywords. Directional derivative, second order Dini upper directional derivative, uniformly convex functions. 1. INTRODUCTION We shall consider the following LC1 problem of unconstrained optimization { } min f ( x) | x ∈ D ⊂ R n , (1) where f : D ⊂ R n → R is a LC1 function on the open convex set D , that means the objective function we want to minimize is continuously differentiable and its gradient ∇f is locally Lipschitzian, i.e. ∇f ( x ) − ∇ f ( y ) ≤ L x − y for x, y ∈ D for some L > 0 . We shall present an iterative algorithm which is based on the algorithms from [1] and [4] for finding an optimal solution to problem (1) generating the sequence of point {xk } of the following form: * This research was supported by Science Fund of Serbia, grant number 1389, through Institute of Mathematics, SANU. 302 N. Đuranović-Miličić / An Algorithm for LC1 Optimization xk +1 = xk + α k sk + α k2 d k , k = 0,1,., sk ≠ 0, d k ≠ 0 (2) where the step-size α k and the directional vectors sk and d k are defined by the particular algorithms. 2. PRELIMINARIES We shall give some preliminaries that will be used for the remainder of the paper. Definition (see [4]). The second order Dini upper directional derivative of the function f ∈ LC1 at xk ∈ R n in the direction d ∈ R n is defined to be [∇f ( xk + λ