TY - UNPB
T1 - Improved worst-case evaluation complexity for potentially rank-deficient nonlinear least-Euclidean-norm problems using higher-order regularized models
AU - Cartis, Coralia
AU - Gould, Nicholas I M
AU - Toint, Philippe
PY - 2015/11/18
Y1 - 2015/11/18
N2 - Given a sufficiently smooth vector-valued function $r(x)$, a local minimizer of $\|r(x)\|_2$ within a closed, non-empty, convex set$\calF$ is sought by modelling $\|r(x)\|^q_2 / q$with a $p$-th order Taylor-series approximation plus a $(p+1)$-st orderregularization term for given even $p$ and some appropriate associated $q$. The resulting algorithm is guaranteed to find a value $\bar{x}$ for which $\|r(\bar{x})\|_2 \leq \epsilon_p$ or $\chi(\bar{x}) \leq \epsilon_d$, for some first-order criticality measure $\chi(x)$ of $\|r(x)\|_2$ within $\calF$,using at most $O(\max\{\max(\epsilon_d,\chi_{\min})^{-(p+1)/p},\max(\epsilon_p,r_{\min})^{-1/2^i}\})$evaluations of $r(x)$ and its derivatives; here $r_{\min}$ and $\chi_{\min} \geq 0$are any lower bounds on $\|r(x)\|_2$ and $\chi(x)$, respectively,and $2^i$ is the highest power of $2$ that divides $p$.
AB - Given a sufficiently smooth vector-valued function $r(x)$, a local minimizer of $\|r(x)\|_2$ within a closed, non-empty, convex set$\calF$ is sought by modelling $\|r(x)\|^q_2 / q$with a $p$-th order Taylor-series approximation plus a $(p+1)$-st orderregularization term for given even $p$ and some appropriate associated $q$. The resulting algorithm is guaranteed to find a value $\bar{x}$ for which $\|r(\bar{x})\|_2 \leq \epsilon_p$ or $\chi(\bar{x}) \leq \epsilon_d$, for some first-order criticality measure $\chi(x)$ of $\|r(x)\|_2$ within $\calF$,using at most $O(\max\{\max(\epsilon_d,\chi_{\min})^{-(p+1)/p},\max(\epsilon_p,r_{\min})^{-1/2^i}\})$evaluations of $r(x)$ and its derivatives; here $r_{\min}$ and $\chi_{\min} \geq 0$are any lower bounds on $\|r(x)\|_2$ and $\chi(x)$, respectively,and $2^i$ is the highest power of $2$ that divides $p$.
M3 - Working paper
VL - 12-2015
T3 - naXys technical report
BT - Improved worst-case evaluation complexity for potentially rank-deficient nonlinear least-Euclidean-norm problems using higher-order regularized models
ER -