Concise complexity analyses for trust region methods

Abstract

Concise complexity analyses are presented for simple trust region algorithms for solving unconstrained optimization problems. In contrast to a traditional trust region algorithm, the algorithms considered in this paper require certain control over the choice of trust region radius after any successful iteration. The analyses highlight the essential algorithm components required to obtain certain complexity bounds. In addition, a new update strategy for the trust region radius is proposed that offers a second-order complexity bound.

An Active Set Trust-Region Method for Bound-Constrained Optimization

“…We prove that the complexity bound O(ε −2 ) holds in both unconstrained and boundconstrained cases. In the unconstrained case, this complexity bound is the same as the known complexity bounds by Curtis [12,13], Grapiglia et al [17], and Gratton et al [18]. In the bound constrained case, our complexity bound is the same as the known complexity results obtained by Cartis et al [7,8] provided that the trust-region algorithms use the quadratic model but with the difference that it is numerically better than them.…”

“…If this critical measure can distinguish numerically and theoretically spurious apparent local minimizers from good local minimizers, it is called a reasonable critical measure. A complexity bound for an iterative bound-constrained optimization method is to use a critical measure to find an upper bound on the number of iterations and terminate at an approximated local minimizer in finite precision arithmetic, e.g., see Birgin et al [2], Cartis et al [7][8][9], Curtis [12,13], Grapiglia et al [17], Gratton et al [18], Nesterov [26], and Nesterov and Polyak [27].…”

Managing public transit during a pandemic: The trade-off between safety and mobility

“…The time complexity of the fixed flow control problem depends on the particular algorithm for the nonlinear optimization. Since we use the trust-region method in this work, assuming a threshold value of on the gradient, the upper bound on the maximum number of iterations is ( Curtis et al, 2018 ). The time complexity analysis of algorithms for flow control with monitoring feedback is not considered for the following reasons.…”

Unconstrained Optimization Methods: Conjugate Gradient Methods and Trust-Region Methods

“…Here, the authors use the modified version of ρ k : where N ∈ N which is originally used by Toint [48]. Something more about trust region methods can be found in [9,18,21,22,54].…”