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### Abstract

We introduce the concept of strong high-order approximate minimizers for
nonconvex optimization problems. These apply in both standard smooth and
composite non-smooth settings, and additionally allow convex or inexpensive
constraints. An adaptive regularization algorithm is then proposed to find such
approximate minimizers. Under suitable Lipschitz continuity assumptions, whenever the feasible set is convex, it is shown that using a model of degree p, this algorithm will find a strong approximate q-th-order minimizer in at most
O(max_{1 \leq j \leq q}\epsilon_j^{-(p+1)/(p-j+1)})
evaluations of the problem’s functions and their derivatives, where \epsilon_j is the j-th order accuracy tolerance; this bound applies when either q = 1 or the problem is not composite with q \leq 2. For general non-composite problems, even when the feasible set is nonconvex, the bound becomes
O(max_{1 \leq j \leq q}\epsilon_j^{-q(p+1)/p})
evaluations. If the problem is composite, and either q > 1 or the
feasible set is not convex, the bound is then
O(max_{1 \leq j \leq q}\epsilon_j^{-(q+1)})
evaluations. These results not only provide, to our knowledge, the first known
bound for (unconstrained or inexpensively-constrained) composite problems for optimality orders exceeding one, but also give the first sharp bounds for high-order strong approximate q-th order minimizers of standard (unconstrained and inexpensively constrained) smooth problems, thereby complementing known results for weak minimizers.

Original language | English |
---|---|

Publisher | Arxiv |

Volume | 2001-10802 |

Publication status | Published - 30 Jan 2020 |

### Keywords

- nonlinear optimisation
- complexity theory
- High-order optimality conditions

## Fingerprint Dive into the research topics of 'Strong Evaluation Complexity Bounds for Arbitrary-Order Optimization of Nonconvex Nonsmooth Composite Functions'. Together they form a unique fingerprint.

## Projects

- 2 Active

## Complexity in nonlinear optimization

TOINT, P., Gould, N. I. M. & Cartis, C.

1/11/08 → …

Project: Research

## Activities

- 1 Invited talk

## Recent results in worst-case evaluation complexity for smooth and non-smooth, exact and inexact, nonconvex optimization

Philippe TOINT (Speaker)

8 May 2020

Activity: Talk or presentation types › Invited talk

## Cite this

Cartis, C., Gould, N., & Toint, P. (2020).

*Strong Evaluation Complexity Bounds for Arbitrary-Order Optimization of Nonconvex Nonsmooth Composite Functions*. Arxiv.