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Modified Piyavskii’s Global One-Dimensional Optimization of a Differentiable Function

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DOI: 10.4236/am.2012.330187    3,298 Downloads   5,673 Views   Citations

ABSTRACT

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 Φ of f and evaluating f at a point where Φ reaches its maximum. We compare the numbers of iterations needed by the modified Piyavskii’s algorithm (nC) to obtain a bounding piece-wise concave function Φ whose maximum is within ε of the globally optimal value foptwith that required by the reference sequential algorithm (nref). The main result is that nC≤ 2nref + 1 and this bound is sharp. We also show that the number of iterations needed by modified Piyavskii’s algorithm to obtain a globally ε-optimal value together with a corresponding point (nB) satisfies nBnref + 1 Lower and upper bounds for nref are obtained as functions of f(x) , ε, M1 and M0 where M0 is a constant defined by M0 = supx∈[a,b] - f’’(x) and M1M0 is an evaluation of M0.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

R. Ellaia, M. Es-Sadek and H. Kasbioui, "Modified Piyavskii’s Global One-Dimensional Optimization of a Differentiable Function," Applied Mathematics, Vol. 3 No. 10A, 2012, pp. 1306-1320. doi: 10.4236/am.2012.330187.

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