Andrew R. Conn

Trust-region methods

Preface 1. Introduction Part I. Preliminaries: 2. Basic Concepts 3. Basic Analysis and Optimality Conditions 4. Basic Linear Algebra 5. Krylov Subspace Methods Part II. Trust-Region Methods for Unconstrained Optimization: 6. Global Convergence of the Basic Algorithm 7.The Trust-Region Subproblem 8. Further Convergence Theory Issues 9. Conditional Models 10. Algorithmic Extensions 11. Nonsmooth Problems Part III. Trust-Region Methods for Constrained Optimization with Convex Constraints: 12. Projection Methods for Convex Constraints 13. Barrier Methods for Inequality Constraints Part IV. Trust-Region Mewthods for General Constained Optimization and Systems of Nonlinear Equations: 14. Penalty-Function Methods 15. Sequential Quadratic Programming Methods 16. Nonlinear Equations and Nonlinear Fitting Part V. Final Considerations: Practicalities Afterword Appendix: A Summary of Assumptions Annotated Bibliography Subject and Notation Index Author Index.

A globally convergent augmented Lagrangian algorithm for optimization with general constraints and simple bounds

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.

Global convergence of a class of trust region algorithms for optimization with simple bounds

This paper extends the known excellent global convergence properties of trust region algorithms for unconstrained optimization to the case where bounds on the variables are present. Weak conditions on the accuracy of the Hessian approximations are considered. It is also shown that, when the strict complementarily condition holds, the proposed algorithms reduce to an unconstrained calculation after finitely many iterations, allowing a fast asymptotic rate of convergence.

Nonlinear programming via an exact penalty function: Asymptotic analysis

In this paper we consider the final stage of a ‘global’ method to solve the nonlinear programming problem. We prove 2-step superlinear convergence. In the process of analyzing this asymptotic behavior, we compare our method (theoretically) to the popular successive quadratic programming approach.

Nonlinear programming via an exact penalty function: Global analysis

In this paper we motivate and describe an algorithm to solve the nonlinear programming problem. The method is based on an exact penalty function and possesses both global and superlinear convergence properties. We establish the global qualities here (the superlinear nature is proven in [7]). The numerical implementation techniques are briefly discussed and preliminary numerical results are given.

A projected Newton method for l p norm location problems

This paper is concerned with the numerical solution of continuous minisum multifacility location problems involving thelp norm, where 1<p<x. This class of problems is potentially difficult to solve because the objective function is not everywhere diflerentiable. After developing conditions that characterize the minimum of the problems under consideration, a second-order algorithm is presented. This algorithm is based on the solution of a finite sequence of linearly constrained subproblems. Descent directions for these subproblems are obtained by projecting the Newton direction onto the corresponding constraint manifold. Univariate minimization is achieved via a specialized linesearch which recognizes the possibility of first derivative discontinuity (and second derivative unboundedness) at points along the search direction. The algorithm, motivated by earlier works of Calamai and Conn, and related to methods recently described by Overton and Dax, is shown to possess both global and quadratic convergence properties.Degeneracy can complicate the numerical solution of the subproblems. This degeneracy is identified, and a method for handling it is outlined.An implementation of the algorithm, that exploits the intrinsic structure of the location problem formulation, is then described along with a discussion of numerical results.

A projection method for the uncapacitated facility location problem

Several algorithms already exist for solving the uncapacitated facility location problem. The most efficient are based upon the solution of the strong linear programming relaxation. The dual of this relaxation has a condensed form which consists of minimizing a certain piecewise linear convex function. This paper presents a new method for solving the uncapacitated facility location problem based upon the exact solution of the condensed dual via orthogonal projections. The amount of work per iteration is of the same order as that of a simplex iteration for a linear program inm variables and constraints, wherem is the number of clients. For comparison, the underlying linear programming dual hasmn + m + n variables andmn +n constraints, wheren is the number of potential locations for the facilities. The method is flexible as it can handle side constraints. In particular, when there is a duality gap, the linear programming formulation can be strengthened by adding cuts. Numerical results for some classical test problems are included.

Linearly Constrained Discrete I 1 Problems

is given, together with an waplementation m Fortran. The algorithm generalizes one given earlier by the same authors for the unconstrained l~ problem.

Second-order conditions for an exact penalty function

In this paper we give first- and second-order conditions to characterize a local minimizer of an exact penalty function. The form of this characterization gives support to the claim that the exact penalty function and the nonlinear programming problem are closely related.In addition, we demonstrate that there exist arguments for the penalty function from which there are no descent directions even though these points are not minimizers.

An exact penalty function for semi-infinite programming

This paper introduces a global approach to the semi-infinite programming problem that is based upon a generalisation of the ℓ1 exact penalty function. The advantages are that the ensuing penalty function is exact and the penalties include all violations. The merit function requires integrals for the penalties, which provides a consistent model for the algorithm. The discretization is a result of the approximate quadrature rather than an a priori aspect of the model.