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

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 (

*n*) to obtain a bounding piece-wise concave function Φ whose maximum is within ε of the globally optimal value_{C}*f*with that required by the reference sequential algorithm (_{opt}*n*). The main result is that_{ref}*n*≤ 2_{C}*n*+ 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 (_{ref}*n*) satisfies_{B}*n*n_{B}_{ref}+ 1 Lower and upper bounds for*n*are obtained as functions of_{ref}*f*(*x*) , ε, M1 and M0 where M0 is a constant defined by*M*_{0}= sup_{x∈[a,b]}-*f’’*(*x*) and*M*_{1}≥*M*_{0}is an evaluation of*M*_{0}.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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