# High-Order Evaluation Complexity for Convexly-Constrained Optimization with Non-Lipschitzian Group Sparsity Terms

Xiaojun Chen, Philippe Toint

Résultats de recherche: Papier de travailArticle de travail

### Résumé

This paper studies high-order evaluation complexity for partially separable
convexly-constrained optimization involving non-Lipschitzian group sparsity
terms in a nonconvex objective function. We propose a partially separable
adaptive regularization algorithm using a p-th order Taylor model and show
that the algorithm can produce an (epsilon,delta)-approximate q-th-order
stationary point at most O(epsilon^{-(p+1)/(p-q+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 non-Lipschitzian 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 partially-separable structure and non-Lipschitzian
group sparsity terms in the objective function may not affect the worst-case
evaluation complexity order.
langue originale Anglais Arxiv 1-27 27 1902.10767 Publié - 28 févr. 2019

### Empreinte digitale

Constrained optimization
Constrained Optimization
Sparsity
Higher Order
Evaluation
Term
Objective function
Euclidean norm
Rotational symmetry
Regularization
Derivatives
Norm
Derivative
Approximation
Model

### Citer ceci

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title = "High-Order Evaluation Complexity for Convexly-Constrained Optimization with Non-Lipschitzian Group Sparsity Terms",
abstract = "This paper studies high-order evaluation complexity for partially separableconvexly-constrained optimization involving non-Lipschitzian group sparsityterms in a nonconvex objective function. We propose a partially separableadaptive regularization algorithm using a p-th order Taylor model and showthat the algorithm can produce an (epsilon,delta)-approximate q-th-orderstationary point at most O(epsilon^{-(p+1)/(p-q+1)}) evaluations of theobjective function and its first p derivatives (whenever they exist). Ourmodel uses the underlying rotational symmetry of the Euclidean norm functionto build a Lipschitzian approximation for the non-Lipschitzian group sparsityterms, which are defined by the group \ell_2-\ell_a norm with a in (0,1). Thenew result shows that the partially-separable structure and non-Lipschitziangroup sparsity terms in the objective function may not affect the worst-caseevaluation complexity order.",
keywords = "complexity theory, nonlinear optimization, non-Lipschitz functions, partially-separable problems, group sparsity, isotropic model, nonlinear optimization, non-Lipschitz functions, partially-separable problems, group sparsity, isotropic model",
author = "Xiaojun Chen and Philippe Toint",
year = "2019",
month = "2",
day = "28",
language = "English",
volume = "1902.10767",
pages = "1--27",
publisher = "Arxiv",
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Arxiv, 2019. p. 1-27.

Résultats de recherche: Papier de travailArticle de travail

TY - UNPB

T1 - High-Order Evaluation Complexity for Convexly-Constrained Optimization with Non-Lipschitzian Group Sparsity Terms

AU - Chen, Xiaojun

AU - Toint, Philippe

PY - 2019/2/28

Y1 - 2019/2/28

N2 - This paper studies high-order evaluation complexity for partially separableconvexly-constrained optimization involving non-Lipschitzian group sparsityterms in a nonconvex objective function. We propose a partially separableadaptive regularization algorithm using a p-th order Taylor model and showthat the algorithm can produce an (epsilon,delta)-approximate q-th-orderstationary point at most O(epsilon^{-(p+1)/(p-q+1)}) evaluations of theobjective function and its first p derivatives (whenever they exist). Ourmodel uses the underlying rotational symmetry of the Euclidean norm functionto build a Lipschitzian approximation for the non-Lipschitzian group sparsityterms, which are defined by the group \ell_2-\ell_a norm with a in (0,1). Thenew result shows that the partially-separable structure and non-Lipschitziangroup sparsity terms in the objective function may not affect the worst-caseevaluation complexity order.

AB - This paper studies high-order evaluation complexity for partially separableconvexly-constrained optimization involving non-Lipschitzian group sparsityterms in a nonconvex objective function. We propose a partially separableadaptive regularization algorithm using a p-th order Taylor model and showthat the algorithm can produce an (epsilon,delta)-approximate q-th-orderstationary point at most O(epsilon^{-(p+1)/(p-q+1)}) evaluations of theobjective function and its first p derivatives (whenever they exist). Ourmodel uses the underlying rotational symmetry of the Euclidean norm functionto build a Lipschitzian approximation for the non-Lipschitzian group sparsityterms, which are defined by the group \ell_2-\ell_a norm with a in (0,1). Thenew result shows that the partially-separable structure and non-Lipschitziangroup sparsity terms in the objective function may not affect the worst-caseevaluation complexity order.

KW - complexity theory, nonlinear optimization, non-Lipschitz functions, partially-separable problems, group sparsity, isotropic model

KW - nonlinear optimization

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KW - partially-separable problems

KW - group sparsity

KW - isotropic model

M3 - Working paper

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BT - High-Order Evaluation Complexity for Convexly-Constrained Optimization with Non-Lipschitzian Group Sparsity Terms

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