A trust region algorithm with a worst-case iteration complexity of \(\mathcal{O}(\epsilon ^{-3/2})\) for nonconvex optimization.

*(English)*Zbl 1360.49020Summary: We propose a trust region algorithm for solving nonconvex smooth optimization problems. For any \(\overline{\epsilon}\in (0,\infty )\), the algorithm requires at most \(\mathcal{O}(\epsilon ^{-3/2})\) iterations, function evaluations, and derivative evaluations to drive the norm of the gradient of the objective function below any \(\epsilon \in (0,\overline{\epsilon}]\). This improves upon the \(\mathcal{O}(\epsilon ^{-2})\) bound known to hold for some other trust region algorithms and matches the \(\mathcal{O}(\epsilon^{-3/2})\) bound for the recently proposed Adaptive Regularisation framework using Cubics, also known as the arc algorithm. Our algorithm, entitled trace, follows a trust region framework, but employs modified step acceptance criteria and a novel trust region update mechanism that allow the algorithm to achieve such a worst-case global complexity bound. Importantly, we prove that our algorithm also attains global and fast local convergence guarantees under similar assumptions as for other trust region algorithms. We also prove a worst-case upper bound on the number of iterations, function evaluations, and derivative evaluations that the algorithm requires to obtain an approximate second-order stationary point.

##### MSC:

49M15 | Newton-type methods |

49M37 | Numerical methods based on nonlinear programming |

58C15 | Implicit function theorems; global Newton methods on manifolds |

65K05 | Numerical mathematical programming methods |

65K10 | Numerical optimization and variational techniques |

65Y20 | Complexity and performance of numerical algorithms |

68Q25 | Analysis of algorithms and problem complexity |

90C30 | Nonlinear programming |

90C60 | Abstract computational complexity for mathematical programming problems |

##### Keywords:

unconstrained optimization; nonlinear optimization; nonconvex optimization; trust region methods; global convergence; local convergence; worst-case iteration complexity; worst-case evaluation complexity##### Software:

GQTPAR
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\textit{F. E. Curtis} et al., Math. Program. 162, No. 1--2 (A), 1--32 (2017; Zbl 1360.49020)

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##### References:

[1] | Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Nonlinear Programming: Theory and Algorithms, 3rd edn. Wiley, London (2006) · Zbl 1140.90040 |

[2] | Bertsekas, D.P.: Nonlinear Programming, 2nd edn. Athena Scientific, Belmont (1999) · Zbl 1015.90077 |

[3] | Cartis, C., Gould, N.I.M., Toint, Ph.L.: On the complexity of steepest descent, Newton’s and regularized Newton’s methods for nonconvex unconstrained optimization problems. SIAM J. Optim. 20(6), 2833-2852 (2010) · Zbl 1211.90225 |

[4] | Cartis, C., Gould, N.I.M., Toint, Ph.L.: Adaptive cubic regularisation methods for unconstrained optimization. Part I: Motivation, convergence and numerical results. Math. Program. 127, 245-295 (2011) · Zbl 1229.90192 |

[5] | Cartis, C., Gould, N.I.M., Toint, Ph.L.: Adaptive cubic regularisation methods for unconstrained optimization. Part II: Worst-case function- and derivative-evaluation complexity. Math. Program. 130, 295-319 (2011) · Zbl 1229.90193 |

[6] | Cartis, C., Gould, N.I.M., Toint. Ph.L.: Optimal Newton-type methods for nonconvex smooth optimization problems. Technical report ERGO 11-009, School of Mathematics, University of Edinburgh (2011) |

[7] | Conn, A.R., Gould, N.I.M., Toint, Ph.L.: Trust-Region Methods. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2000) · Zbl 0958.65071 |

[8] | Dennis, J.E., Schnabel, R.B.: Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1996) · Zbl 0847.65038 |

[9] | Griewank, A.: The modification of Newton’s method for unconstrained optimization by bounding cubic terms. Technical report NA/12, Department of Applied Mathematics and Theoretical Physics, University of Cambridge (1981) |

[10] | Griva, I., Nash, S.G., Sofer, A.: Linear and Nonlinear Optimization, 2nd edn. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2008) · Zbl 1159.90002 |

[11] | Moré, JJ; Sorensen, DC, Computing a trust region step, SIAM J. Sci. Stat. Comput., 4, 553-572, (1983) · Zbl 0551.65042 |

[12] | Nesterov, Yu; Polyak, BT, Cubic regularization of newton’s method and its global performance, Math. Program., 108, 117-205, (2006) · Zbl 1142.90500 |

[13] | Nesterov, Y.: Introductory Lectures on Convex Optimization, vol. 87. Springer, Berlin (2004) · Zbl 1086.90045 |

[14] | Nocedal, J., Wright, S.J.: Numerical Optimization. Springer Series in Operations Research and Financial Engineering, 2nd edn. Springer, Berlin (2006) |

[15] | Ruszczynski, A.: Nonlinear Optimization. Princeton University Press, Princeton (2006) · Zbl 1108.90001 |

[16] | Weiser, M; Deuflhard, P; Erdmann, B, Affine conjugate adaptive Newton methods for nonlinear elastomechanics, Optim. Methods Softw., 22, 413-431, (2007) · Zbl 1128.74007 |

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