Worst-case evaluation complexity for unconstrained nonlinear optimization using high-order regularized models |
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| Authors: | E G Birgin J L Gardenghi J M Martínez S A Santos Ph L Toint |
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| Institution: | 1.Department of Computer Science, Institute of Mathematics and Statistics,University of S?o Paulo,S?o Paulo,Brazil;2.Department of Applied Mathematics, Institute of Mathematics, Statistics, and Scientific Computing,University of Campinas,Campinas,Brazil;3.Namur Center for Complex Systems (naXys) and Department of Mathematics,University of Namur,Namur,Belgium |
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| Abstract: | The worst-case evaluation complexity for smooth (possibly nonconvex) unconstrained optimization is considered. It is shown that, if one is willing to use derivatives of the objective function up to order p (for \(p\ge 1\)) and to assume Lipschitz continuity of the p-th derivative, then an \(\epsilon \)-approximate first-order critical point can be computed in at most \(O(\epsilon ^{-(p+1)/p})\) evaluations of the problem’s objective function and its derivatives. This generalizes and subsumes results known for \(p=1\) and \(p=2\). |
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