Partially separable convexly-constrained optimization with non-Lipschitzian singularities and its complexity

Xiaojun Chen, Philippe Toint, Hong Wang

Research output: Contribution to journalArticle

Abstract

An adaptive regularization algorithm using high-order models is proposed for partially separable convexly constrained nonlinear optimization problems whose objective function contains non-Lipschitzian ℓq-norm regularization terms for q ∈ (0,1). It is shown that the algorithm using an p-th order Taylor model for p odd needs in general at most O (ǫ−(p+1)/p) evaluations of the objective function and its derivatives (at points where they are defined) to produce an ǫ-approximate first-order critical point. This result is obtained either with Taylor models at the price of requiring the feasible set to be ’kernel-centered’ (which includes bound constraints and many other cases of interest ), or for non-Lipschitz models, at the price of passing the difficulty to the computation of the step. Since this complexity bound is identical in order to that already known for purely Lipschitzian minimization subject to convex constraints [9], the new result shows that introducing non-Lipschitzian singularities in the objective function may not affect the worst-case evaluation complexity order. The result also shows that using the problem’s partially separable structure (if present) does not affect complexity order either. A final (worse) complexity bound is derived for the case where Taylor models are used with a general convex feasible set.
Original languageEnglish
Pages (from-to)874-903
Number of pages30
JournalSIAM Journal on Optimization
Volume29
Issue number1
Publication statusPublished - 15 Apr 2019

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Constrained optimization
Constrained Optimization
Singularity
Objective function
Regularization
Bound Constraints
Non-Lipschitz
Convex Constraints
Model
Evaluation
Adaptive algorithms
Nonlinear Optimization
Nonlinear Problem
Critical point
Odd
Higher Order
kernel
Optimization Problem
First-order
Derivatives

Keywords

  • Non_Lipschitz optimization
  • Complexity theory
  • partial separability

Cite this

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title = "Partially separable convexly-constrained optimization with non-Lipschitzian singularities and its complexity",
abstract = "An adaptive regularization algorithm using high-order models is proposed for partially separable convexly constrained nonlinear optimization problems whose objective function contains non-Lipschitzian ℓq-norm regularization terms for q ∈ (0,1). It is shown that the algorithm using an p-th order Taylor model for p odd needs in general at most O (ǫ−(p+1)/p) evaluations of the objective function and its derivatives (at points where they are defined) to produce an ǫ-approximate first-order critical point. This result is obtained either with Taylor models at the price of requiring the feasible set to be ’kernel-centered’ (which includes bound constraints and many other cases of interest ), or for non-Lipschitz models, at the price of passing the difficulty to the computation of the step. Since this complexity bound is identical in order to that already known for purely Lipschitzian minimization subject to convex constraints [9], the new result shows that introducing non-Lipschitzian singularities in the objective function may not affect the worst-case evaluation complexity order. The result also shows that using the problem’s partially separable structure (if present) does not affect complexity order either. A final (worse) complexity bound is derived for the case where Taylor models are used with a general convex feasible set.",
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Partially separable convexly-constrained optimization with non-Lipschitzian singularities and its complexity. / Chen, Xiaojun; Toint, Philippe; Wang, Hong.

In: SIAM Journal on Optimization, Vol. 29, No. 1, 15.04.2019, p. 874-903.

Research output: Contribution to journalArticle

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AB - An adaptive regularization algorithm using high-order models is proposed for partially separable convexly constrained nonlinear optimization problems whose objective function contains non-Lipschitzian ℓq-norm regularization terms for q ∈ (0,1). It is shown that the algorithm using an p-th order Taylor model for p odd needs in general at most O (ǫ−(p+1)/p) evaluations of the objective function and its derivatives (at points where they are defined) to produce an ǫ-approximate first-order critical point. This result is obtained either with Taylor models at the price of requiring the feasible set to be ’kernel-centered’ (which includes bound constraints and many other cases of interest ), or for non-Lipschitz models, at the price of passing the difficulty to the computation of the step. Since this complexity bound is identical in order to that already known for purely Lipschitzian minimization subject to convex constraints [9], the new result shows that introducing non-Lipschitzian singularities in the objective function may not affect the worst-case evaluation complexity order. The result also shows that using the problem’s partially separable structure (if present) does not affect complexity order either. A final (worse) complexity bound is derived for the case where Taylor models are used with a general convex feasible set.

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