No abstract

The global and local convergence properties of a class of augmented Lagrangian methods for solving nonlinear programming problems are considered. In such methods, simple bound constraints are treated separately from more general constraints and the stopping rules for the inner minimization algorithm have this in mind. Global convergence is proved, and it is established that a potentially troublesome penalty parameter is bounded away from zero.

An Adaptive Regularisation framework using Cubics (ARC) was proposed for unconstrained optimization and analysed in Cartis, Gould & Toint (Part I, 2007). In this companion paper, we further the analysis by providing worst-case global iteration complexity bounds for ARC and a second-order variant to achieve approximate first-order, and for the latter even second-order, criticality of the iterates. In particular, the second-order ARC algorithm requires at most O(ǫ −3/2 ) iterations to drive the objective's gradient below the desired accuracy ǫ, and O(ǫ −3 ), to reach approximate nonnegative curvature in a subspace. The orders of these bounds match those proved by Nesterov & Polyak (Math. Programming 108(1), 2006, pp 177-205) for their Algorithm 3.3 which minimizes the cubic model globally on each iteration. Our approach is more general, and relevant to practical (large-scale) calculations, as ARC allows the cubic model to be solved only approximately and may employ approximate Hessians.

An Adaptive Regularisation framework using Cubics (ARC) was proposed for unconstrained optimization and analysed in Cartis, Gould & Toint (Part I, 2007). In this companion paper, we further the analysis by providing worst-case global iteration complexity bounds for ARC and a second-order variant to achieve approximate first-order, and for the latter even second-order, criticality of the iterates. In particular, the second-order ARC algorithm requires at most O(ǫ −3/2) iterations to drive the objective's gradient below the desired accuracy ǫ, and O(ǫ −3), to reach approximate nonnegative curvature in a subspace. The orders of these bounds match those proved by Nesterov & Polyak (Math. Programming 108(1), 2006, pp 177-205) for their Algorithm 3.3 which minimizes the cubic model globally on each iteration. Our approach is more general, and relevant to practical (large-scale) calculations , as ARC allows the cubic model to be solved only approximately and may employ approximate Hessians.

The purpose of this article is to discuss the scope and functionality of a versatile environment for testing small- and large-scale nonlinear optimization algorithms. Although many of these facilities were originally produced by the authors in conjunction with the software package LANCELOT, we believe that they will be useful in their own right and should be available to researchers for their development of optimization software. The tools can be obtained by anonymous ftp from a number of sources and may, in many cases, be installed automatically. The scope of a major collection of test problems written in the standard input format (SIF) used by the LANCELOT software package is described. Recognizing that most software was not written with the SIF in mind, we provide tools to assist in building an interface between this input format and other optimization packages. These tools provide a link between the SIF and a number of existing packages, including MINOS and OSL. Additionally, as each problem includes a specific classification that is designed to be useful in identifying particular classes of problems, facilities are provided to build and manage a database of this information. There is a Unix and C shell bias to many of the descriptions in the article, since, for the sake of simplicity, we do not illustrate everything in its fullest generality. We trust that the majority of potential users are sufficiently familiar with Unix that these examples will not lead to undue confusion.