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Abstract
This paper studies highorder evaluation complexity for partially separable
convexlyconstrained optimization involving nonLipschitzian group sparsity
terms in a nonconvex objective function. We propose a partially separable
adaptive regularization algorithm using a pth order Taylor model and show
that the algorithm can produce an (epsilon,delta)approximate qthorder
stationary point at most O(epsilon^{(p+1)/(pq+1)}) evaluations of the
objective function and its first p derivatives (whenever they exist). Our
model uses the underlying rotational symmetry of the Euclidean norm function
to build a Lipschitzian approximation for the nonLipschitzian group sparsity
terms, which are defined by the group \ell_2\ell_a norm with a in (0,1). The
new result shows that the partiallyseparable structure and nonLipschitzian
group sparsity terms in the objective function may not affect the worstcase
evaluation complexity order.
convexlyconstrained optimization involving nonLipschitzian group sparsity
terms in a nonconvex objective function. We propose a partially separable
adaptive regularization algorithm using a pth order Taylor model and show
that the algorithm can produce an (epsilon,delta)approximate qthorder
stationary point at most O(epsilon^{(p+1)/(pq+1)}) evaluations of the
objective function and its first p derivatives (whenever they exist). Our
model uses the underlying rotational symmetry of the Euclidean norm function
to build a Lipschitzian approximation for the nonLipschitzian group sparsity
terms, which are defined by the group \ell_2\ell_a norm with a in (0,1). The
new result shows that the partiallyseparable structure and nonLipschitzian
group sparsity terms in the objective function may not affect the worstcase
evaluation complexity order.
Original language  English 

Pages (fromto)  127 
Number of pages  27 
Journal  Mathematical Programming 
Volume  1902.10767 
DOIs  
Publication status  Accepted/In press  1 Jan 2020 
Keywords
 complexity theory, nonlinear optimization, nonLipschitz functions, partiallyseparable problems, group sparsity, isotropic model
 nonlinear optimization
 nonLipschitz functions
 partiallyseparable problems
 group sparsity
 isotropic model
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Projects
 1 Active

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

Recent results in worstcase evaluation complexity for smooth and nonsmooth, exact and inexact, nonconvex optimization
Philippe TOINT (Speaker)
8 May 2020Activity: Talk or presentation types › Invited talk

Departement of Applied Mathematics, Polytechnic University of Hong Kong
Philippe Toint (Visiting researcher)
15 Nov 2018 → 15 Dec 2018Activity: Visiting an external institution types › Visiting an external academic institution