%0 Journal Article
%T Modified Piyavskii¡¯s Global One-Dimensional Optimization of a Differentiable Function
%A Rachid Ellaia
%A Mohamed Zeriab Es-Sadek
%A Hasnaa Kasbioui
%J Applied Mathematics
%P 1306-1320
%@ 2152-7393
%D 2012
%I Scientific Research Publishing
%R 10.4236/am.2012.330187
%X Piyavskii¡¯s algorithm maximizes a univariate function satisfying a Lipschitz condition. We propose a modified Piyavskii¡¯s sequential algorithm which maximizes a univariate differentiable function f by iteratively constructing an upper bounding piece-wise concave function ¦µ of f and evaluating f at a point where ¦µ reaches its maximum. We compare the numbers of iterations needed by the modified Piyavskii¡¯s algorithm (<i>n<sub>C</sub></i>) to obtain a bounding piece-wise concave function ¦µ whose maximum is within ¦Å of the globally optimal value <i>f<sub>opt</sub></i>with that required by the reference sequential algorithm (<i>n<sub>ref</sub></i>). The main result is that <i>n<sub>C</sub></i>¡Ü 2<i>n<sub>ref</sub></i> + 1 and this bound is sharp. We also show that the number of iterations needed by modified Piyavskii¡¯s algorithm to obtain a globally ¦Å-optimal value together with a corresponding point (<i>n<sub>B</sub></i>) satisfies <i>n<sub>B</sub></i>n<sub>ref</sub> + 1 Lower and upper bounds for <i>n<sub>ref</sub></i> are obtained as functions of <i>f</i>(<i>x</i>) , ¦Å, M1 and M0 where M0 is a constant defined by <i>M</i><sub>0</sub> = sup<sub>x¡Ê[a,b]</sub> - <i>f¡¯¡¯</i>(<i>x</i>) and <i>M</i><sub>1</sub> ¡İ <i>M</i><sub>0</sub> is an evaluation of <i>M</i><sub>0</sub>.
%K Global Optimization
%K Piyavskii¡¯s Algorithm
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=24105