### 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 language | English |
---|---|

Pages (from-to) | 874-903 |

Number of pages | 30 |

Journal | SIAM Journal on Optimization |

Volume | 29 |

Issue number | 1 |

DOIs | |

Publication status | Published - 15 Apr 2019 |

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### Keywords

- Non_Lipschitz optimization
- Complexity theory
- partial separability

### Cite this

*SIAM Journal on Optimization*,

*29*(1), 874-903. https://doi.org/10.1137/18M1166511

}

*SIAM Journal on Optimization*, vol. 29, no. 1, pp. 874-903. https://doi.org/10.1137/18M1166511

**Complexity of partially separable convexly constrained optimization with non-Lipschitzian singularities.** / Chen, Xiaojun; Toint, Philippe; Wang, Hong.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Complexity of partially separable convexly constrained optimization with non-Lipschitzian singularities

AU - Chen, Xiaojun

AU - Toint, Philippe

AU - Wang, Hong

PY - 2019/4/15

Y1 - 2019/4/15

N2 - 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.

AB - 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.

KW - Non_Lipschitz optimization

KW - Complexity theory

KW - partial separability

UR - http://www.scopus.com/inward/record.url?scp=85065443105&partnerID=8YFLogxK

U2 - 10.1137/18M1166511

DO - 10.1137/18M1166511

M3 - Article

VL - 29

SP - 874

EP - 903

JO - SIAM Journal on Optimization

JF - SIAM Journal on Optimization

SN - 1052-6234

IS - 1

ER -