Complexity of partially separable convexly constrained optimization with non-lipschitzian singularities

Xiaojun Chen, Philippe Toint, Hong Wang

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Abstract

An adaptive regularization algorithm using high-order models is proposed for solving partially separable convexly constrained nonlinear optimization problems whose objective function contains non-Lipschitzian 'q-norm regularization terms for q ϵ (0; 1). It is shown that the algorithm using a pth-order Taylor model for p odd needs in general at most O(ϵ -(p+1)=p) evaluations of the objective function and its derivatives (at points where they are defined) to produce an ϵ-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 with 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 [C. Cartis, N. I. M. Gould, and Ph. L. Toint, IMA J. Numer. Anal., 32 (2012), pp. 1662-1695], 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 the 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
DOIs
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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Complexity of partially separable convexly constrained optimization with non-lipschitzian singularities. / 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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