Sensitivity of trust-region algorithms to their parameters

In this paper, we examine the sensitivity of trust-region algorithms on the parameters related to the step acceptance and update of the trust region. We show, in the context of unconstrained programming, that the numerical efficiency of these algorithms can easily be improved by choosing appropriate parameters. Recommended ranges of values for these parameters are exhibited on the basis of extensive numerical tests. MSC classification: 65K05, 90C26, 90C30     

Nonlinear programming without a penalty function or a filter

A new method is introduced for solving equality constrained nonlinear optimization problems. This method does not use a penalty function, nor a filter, and yet can be proved to be globally convergent to first-order stationary points. It uses different trust-regions to cope with the nonlinearities of the objective function and the constraints, and allows inexact SQP steps that do not lie exactly in the nullspace of the local Jacobian. Preliminary numerical experiments on CUTEr problems indicate that the method performs well.      

On the complexity of finding first-order critical points in constrained nonlinear optimization

The complexity of finding $\epsilon $ -approximate first-order critical points for the general smooth constrained optimization problem is shown to be no worse that $O(\epsilon ^{-2})$ in terms of function and constraints evaluations. This result is obtained by analyzing the worst-case behaviour of a first-order short-step homotopy algorithm consisting of a feasibility phase followed by an optimization phase, and requires minimal assumptions on the objective function. Since a bound of the same order is known to be valid for the unconstrained case, this leads to the conclusion that the presence of possibly nonlinear/nonconvex inequality/equality constraints is irrelevant for this bound to apply.     

Preconditioning and globalizing conjugate gradients in dual space for quadratically penalized nonlinear-least squares problems

When solving nonlinear least-squares problems, it is often useful to regularize the problem using a quadratic term, a practice which is especially common in applications arising in inverse calculations. A solution method derived from a trust-region Gauss-Newton algorithm is analyzed for such applications, where, contrary to the standard algorithm, the least-squares subproblem solved at each iteration of the method is rewritten as a quadratic minimization subject to linear equality constraints. This allows the exploitation of duality properties of the associated linearized problems. This paper considers a recent conjugate-gradient-like method which performs the quadratic minimization in the dual space and produces, in exact arithmetic, the same iterates as those produced by a standard conjugate-gradients method in the primal space. This dual algorithm is computationally interesting whenever the dimension of the dual space is significantly smaller than that of the primal space, yielding gains in terms of both memory usage and computational cost. The relation between this dual space solver and PSAS (Physical-space Statistical Analysis System), another well-known dual space technique used in data assimilation problems, is explained. The use of an effective preconditioning technique is proposed and refined convergence bounds derived, which results in a practical solution method. Finally, stopping rules adequate for a trust-region solver are proposed in the dual space, providing iterates that are equivalent to those obtained with a Steihaug-Toint truncated conjugate-gradient method in the primal space.     

An interior-point trust-funnel algorithm for nonlinear optimization

We present an interior-point trust-funnel algorithm for solving large-scale nonlinear optimization problems. The method is based on an approach proposed by Gould and Toint (Math Prog 122(1):155–196, 2010) that focused on solving equality constrained problems. Our method is similar in that it achieves global convergence guarantees by combining a trust-region methodology with a funnel mechanism, but has the additional capability of being able to solve problems with both equality and inequality constraints. The prominent features of our algorithm are that (i) the subproblems that define each search direction may be solved with matrix-free methods so that derivative matrices need not be formed or factorized so long as matrix-vector products with them can be performed; (ii) the subproblems may be solved approximately in all iterations; (iii) in certain situations, the computed search directions represent inexact sequential quadratic optimization steps, which may be desirable for fast local convergence; (iv) criticality measures for feasibility and optimality aid in determining whether only a subset of computations need to be performed during a given iteration; and (v) no merit function or filter is needed to ensure global convergence.     

Adaptive cubic regularisation methods for unconstrained optimization. Part I: motivation,convergence and numerical results

An Adaptive Regularisation algorithm using Cubics (ARC) is proposed for unconstrained optimization, generalizing at the same time an unpublished method due to Griewank (Technical Report NA/12, 1981, DAMTP, University of Cambridge), an algorithm by Nesterov and Polyak (Math Program 108(1):177–205, 2006) and a proposal by Weiser et al. (Optim Methods Softw 22(3):413–431, 2007). At each iteration of our approach, an approximate global minimizer of a local cubic regularisation of the objective function is determined, and this ensures a significant improvement in the objective so long as the Hessian of the objective is locally Lipschitz continuous. The new method uses an adaptive estimation of the local Lipschitz constant and approximations to the global model-minimizer which remain computationally-viable even for large-scale problems. We show that the excellent global and local convergence properties obtained by Nesterov and Polyak are retained, and sometimes extended to a wider class of problems, by our ARC approach. Preliminary numerical experiments with small-scale test problems from the CUTEr set show encouraging performance of the ARC algorithm when compared to a basic trust-region implementation.     

Stopping rules and backward error analysis for bound-constrained optimization

Termination criteria for the iterative solution of bound-constrained optimization problems are examined in the light of backward error analysis. It is shown that the problem of determining a suitable perturbation on the problem’s data corresponding to the definition of the backward error is analytically solvable under mild assumptions. Moreover, a link between existing termination criteria and this solution is clarified, indicating that some standard measures of criticality may be interpreted in the sense of backward error analysis. The backward error problem is finally considered from the multicriteria optimization point of view and some numerical illustration is provided.     

Fast regularized linear sampling for inverse scattering problems

A new numerical procedure is proposed for the reconstruction of the shape and volume of unknown objects from measurements of their radiation in the far field. This procedure is a variant and the linear sampling method has a very acceptable computational load and is fully automated. It is based on exploiting an iteratively computed truncated singular‐value decomposition and heuristics to extract the desired signal from the background noise. Its performance on a battery of examples of different types is shown to be promising. Copyright © 2010 John Wiley & Sons, Ltd.     

Minimizing convex quadratics with variable precision conjugate gradients

We investigate the method of conjugate gradients, exploiting inaccurate matrix‐vector products, for the solution of convex quadratic optimization problems. Theoretical performance bounds are derived, and the necessary quantities occurring in the theoretical bounds estimated, leading to a practical algorithm. Numerical experiments suggest that this approach has significant potential, including in the steadily more important context of multiprecision computations.     

On the Number of Inner Iterations Per Outer Iteration of a Globally Convergent Algorithm for Optimization with General Nonlinear Inequality Constraints and Simple Bounds

This paper considers the number of inner iterations required per outeriteration for the algorithm proposed by Conn et al.[9]. We show that asymptotically, under suitable reasonable assumptions, a single inner iteration suffices.