Worst-case evaluation complexity for unconstrained nonlinear optimization using high-order regularized models

FAPESP - FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULOCNPQ - CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICOThe 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 >= 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.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 >= 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 = 2.1631-2359368FAPESP - FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULOCNPQ - CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICOFAPESP - FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULOCNPQ - CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICO2010/10133-0; 2013/03447-6; 2013/05475-7; 2013/07375-0; 2013/23494-9304032/2010-7; 309517/2014-1; 303750/2014-6; 490326/2013-