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Abstract
A stochastic adaptive regularization algorithm allowing random noise in
derivatives and inexact function values is proposed for computing strong
approximate minimizers of any order for inexpensively constrained smooth
optimization problems. For an objective function with Lipschitz continuous
pth derivative in a convex neighbourhood of the feasible set and
given an arbitrary optimality order q, it is shown that this algorithm
will, in expectation, compute such a point in at most
O( ( \min_{1<=j<=q} \epsilon_j )^{(p+1)/(pq+1)} )
inexact evaluations of f and its derivatives whenever q=1 and the
feasible set is convex, or q=2 and the problem is unconstrained, where
\epsilon_j is the tolerance for jth order accuracy. This bound becomes
at most
O( ( \min_{1<=j<=q} \epsilon_j )^{q(p+1)/p} )
inexact evaluations in the other cases if all derivatives are Lipschitz
continuous. Moreover these bounds are sharp in the order of the accuracy
tolerances.
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
