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On a second order step-size algorithm

In this paper we present a modification of the second-order step-size algorithm. This modification is based on the so called "forcing functions". It is proved that this modified algorithm is well-defined. It is also proved that every point of accumulation of the sequence generated by this algorithm is a second-order point of the nonlinear programming problem. | Yugoslav Journal of Operations Research 12 (2002), Number 1, 121-127 ON A SECONDSTEP EPSECOND-ORDER ST EP-SIZE ALGORITHM* Nada I. DJURANOVI]-MILI^I] Department of Mathematics Faculty of Technology and Metallurgy University of Belgrade, Belgrade, Yugoslavia nmilicic@elab.tmf.bg.ac.yu Abstract: In this paper we present a modification of the second-order step-size algorithm. This modification is based on the so called "forcing functions". It is proved that this modified algorithm is well-defined. It is also proved that every point of accumulation of the sequence generated by this algorithm is a second-order point of the nonlinear programming problem. Two different convergence proofs are given having in mind two interpretations of the presented algorithm. Keywords: Forcing function, step-size algorithm, second-order conditions. 1. INTRODUCTION We are concerned with the following problem of the unconstrained optimization: min{ϕ ( x) | x ∈ D} (1) where ϕ : D ⊂ Rn → R is a twicecontinuously differentiable function on an open set D. We consider iterative algorithms to find an optimal solution to problem (1) generating sequences of points {xk} of the following form: xk+1 = xk + α ksk + β kdk , k = 0,1,. , sk , dk ≠ 0, ∇ϕ ( xk ), sk ≤ 0, and the steps α k and β k are defined by a particular step-size algorithm. * This research was supported by Science Fund of Serbia, grant number 04M03, through Institute of Mathematics, SANU. AMS Mathematics Subject Classification (1991): 90C30 (2) (3) 122 N. Djuranovi}-Mili~i} / On a Second-Order Step-Size Algorithm Before we present the modified algorithm, we shall define the original secondorder step-size algorithm. The original Mc Cormick-Armijo's second order step-size algorithm [4] defines α k in the following way: α k > 0 is a number satisfying α k = 2 − i ( k) , where i( k) is the smallest integer from i = 0,1,. , such that xk+1 = xk + 2−i( k) sk + 2 − i( k) 2 d ∈D k and 1 ϕ ( xk ) − ϕ ( xk+1 ) ≥ γ − ∇ϕ ( xk ),