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Abstract
A regularization algorithm allowing random noise in derivatives and inexact
function values is proposed for computing approximate local critical points
of any order for smooth unconstrained optimization problems. For an
objective function with Lipschitz continuous pth derivative and given an
arbitrary optimality order q <= p, it is shown that this algorithm will, in
expectation, compute such a point in at most
O((min_{j=1,...,q}epsilon_j)^{(p+1)/(pq+1)})
inexact evaluations of f and its derivatives whenever q in {1,2}, where
\epsilon_j is the tolerance for jth order accuracy. This bound becomes at most
O((min_{j=1,...,q}epsilon_j)^{(q(p+1))/(p)})
inexact evaluations if q>2 and all derivatives are Lipschitz
continuous. Moreover these bounds are sharp in the order of the accuracy
tolerances. An extension to convexly constrained problems is also outlined.
function values is proposed for computing approximate local critical points
of any order for smooth unconstrained optimization problems. For an
objective function with Lipschitz continuous pth derivative and given an
arbitrary optimality order q <= p, it is shown that this algorithm will, in
expectation, compute such a point in at most
O((min_{j=1,...,q}epsilon_j)^{(p+1)/(pq+1)})
inexact evaluations of f and its derivatives whenever q in {1,2}, where
\epsilon_j is the tolerance for jth order accuracy. This bound becomes at most
O((min_{j=1,...,q}epsilon_j)^{(q(p+1))/(p)})
inexact evaluations if q>2 and all derivatives are Lipschitz
continuous. Moreover these bounds are sharp in the order of the accuracy
tolerances. An extension to convexly constrained problems is also outlined.
Original language  English 

Publisher  Arxiv 
Number of pages  22 
Volume  2005.04639 
Publication status  Published  13 May 2020 
Keywords
 evaluation complexity
 regularization methods
 lnexact functions and derivatives
 stochastic analysis
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Projects
 2 Active

Complexity in nonlinear optimization
TOINT, P., Gould, N. I. M. & Cartis, C.
1/11/08 → …
Project: Research
