Abstract. This paper concerns large-scale general (nonconvex) nonlinear programming when first and second derivatives of the objective and constraint functions are available. A method is proposed that is based on finding an approximate solution of a sequence of unconstrained subproblems parameterized by a scalar parameter. The objective function of each unconstrained subproblem is an augmented penalty-barrier function that involves both primal and dual variables. Each subproblem is solved with a modified Newton method that generates search directions from a primal-dual system similar to that proposed for interior methods. The augmented penalty-barrier function may be interpreted as a merit function for values of the primal and dual variables. An inertia-controlling symmetric indefinite factorization is used to provide descent directions and directions of negative curvature for the augmented penalty-barrier merit function. A method suitable for large problems can be obtained by providing a version of this factorization that will treat large sparse indefinite systems.

Abstract. Interior methods are an omnipresent, conspicuous feature of the constrained optimization landscape today, but it was not always so. Primarily in the form of barrier methods, interior-point techniques were popular during the 1960s for solving nonlinearly constrained problems. However, their use for linear programming was not even contemplated because of the total dominance of the simplex method. Vague but continuing anxiety about barrier methods eventually led to their abandonment in favor of newly emerging, apparently more efficient alternatives such as augmented Lagrangian and sequential quadratic programming methods. By the early 1980s, barrier methods were almost without exception regarded as a closed chapter in the history of optimization. This picture changed dramatically with Karmarkar’s widely publicized announcement in 1984 of a fast polynomial-time interior method for linear programming; in 1985, a formal connection was established between his method and classical barrier methods. Since then, interior methods have advanced so far, so fast, that their influence has transformed both the theory and practice of constrained optimization. This article provides a condensed, selective look at classical material and recent research about interior methods for nonlinearly constrained optimization.

. We present a new trust region algorithm for solving nonlinear equality constrained optimization problems. At each iterate a change of variables is performed to improve the ability of the algorithm to follow the constraint level sets. The algorithm employs L 2 penalty functions for obtaining global convergence. Under certain assumptions we prove that this algorithm globally converges to a point satisfying the second order necessary optimality conditions; the local convergence rate is quadratic. Results of preliminary numerical experiments are presented. 1. Introduction. We consider the equality constrained optimization problem minimize f(x) subject to c(x) = 0 (1:1) where x 2 ! n and f : ! n ! !, and c : ! n ! ! m are smooth nonlinear functions. Problem (1.1) is often solved by successive quadratic programming (SQP) methods. At a current point x k 2 ! n , SQP methods determine a search direction d k by solving a quadratic programming problem minimize rf(x k ) T d + 1 2 ...

Consider a full-rank weighted least-squares problem in which the weight matrix is highly ill-conditioned. Because of the ill-conditioning, standard methods for solving least-squares problems, QR factorization and the nullspace method for example, break down. G. W. Stewart established a norm bound for such a system of equations, indicating that it may be possible to find an algorithm that gives an accurate solution. S. A. Vavasis proposed a new definition of stability that is based on this result. He also defined the NSH algorithm for solving this least-squares problem and showed that it satisfies his definition of stability. In this paper, we propose a complete orthogonal decomposition algorithm to solve this problem and show that it is also stable. This new algorithm is simpler and more efficient than the NSH method. 1 Introduction We consider solving the problem min y2R n kD \Gamma1=2 (Ay \Gamma b) k (1) for y, where D is a symmetric positive definite m \Theta m matrix, A is an ...

Abstract. In the Newton/log-barrier method, Newton steps are taken for the log-barrier function for a xed value of the barrier parameter until a certain convergence criterion is satis ed. The barrier parameter is then decreased and the Newton process is repeated. A naive analysis indicates that Newton's method does not exhibit superlinear convergence to the minimizer of each instance of the log-barrier function until it reaches a very small neighborhood of the minimizer. By partitioning according to the subspace of active constraint gradients, however, we show that this neighborhood is actually quite large, thus explaining why reasonably fast local convergence can be attained in practice. Moreover, we show that the overall convergence rate of the Newton/log-barrier algorithm is superlinear in the number of function/derivative evaluations, provided that the nonlinear program is formulated with a linear objective and that the schedule for decreasing the barrier parameter is related in a certain way to the convergence criterion for each Newton process. 1.

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. We examine the sequence of local minimizers of the log-barrier function for a nonlinear program near a solution at which second-ordersufficient conditions and the MangasarianFromovitz constraint qualifications are satisfied, but the active constraint gradients are not necessarily linearly independent. When a strict complementarity condition is satisfied, we show uniqueness of the local minimizer of the barrier function in the vicinity of the nonlinear program solution, and obtain a semi-explicit characterization of this point. When strict complementarity does not hold, we obtain several other interesting characterizations, in particular, an estimate of the distance between the minimizers of the barrier function and the nonlinear program in terms of the barrier parameter, and a result about the direction of approach of the sequence of minimizers of the barrier function to the nonlinear programming solution. 1. Introduction We consider the nonlinear programming problem min f(x) subjec...

. Inequality constrained minimization problems are often solved by considering a sequence of parameterized barrier functions. Each barrier function is approximately minimized and the relevant parameters subsequently adjusted. It is common for the estimated solution to one barrier function problem to be used as a starting estimate for the next. However, this has unfortunate repercussions for the standard Newton-like methods applied to the barrier subproblem. In this note, we consider a class of alternative Newton methods which attempt to avoid such difficulties. Such schemes have already proved of use in the Harwell Subroutine Library quadratic programming codes VE14 and VE19. 1 IBM T.J. Watson Research Center, P.O.Box 218, Yorktown Heights, NY 10598, USA Email : arconn@watson.ibm.com 2 CERFACS, 42 Avenue Gustave Coriolis, 31057 Toulouse Cedex, France, EC Email : gould@cerfacs.fr or nimg@directory.rl.ac.uk 3 Department of Mathematics, Facult'es Universitaires ND de la Paix, 61, rue...

We show that the effects of finite-precision arithmetic in forming and solving the linear system that arises at each iteration of primal-dual interior-point algorithms for nonlinear programming are benign. When we replace the standard assumption that the active constraint gradients are independentby the weaker Mangasarian-Fromovitz constraint qualifiation, rapid convergence usually is attainable, even when cancellation and roundoff errors occur during the calculations. In deriving our main results, we proveakey technical result about the size of the exact primal-dual step. This result can be used to modify existing analysis of primal-dual interior-point methods for convex programming, making it possible to extend the superlinear local convergence results to the nonconvex case.

Preconditioned conjugate-gradient methods are proposed for solving the ill-conditioned linear systems which arise in penalty and barrier methods for nonlinear minimization. The preconditioners are chosen so as to isolate the dominant cause of ill conditioning. The methods are stablized using a restricted form of iterative refinement. Numerical results illustrate the approaches considered. 1 Email : n.gould@rl.ac.uk 2 Current reports available from "http://www.rl.ac.uk/departments/ccd/numerical/reports/reports.html". Department for Computation and Information Atlas Centre Rutherford Appleton Laboratory Oxfordshire OX11 0QX August 26, 1998. 1 INTRODUCTION 1 1 Introduction Let A and H be, respectively, full-rank m by n (m n) and symmetric n by n real matrices. Suppose furthermore that any nonzero coefficients in this data are modest, that is the data is O(1). (1) We consider the iterative solution of the linear system (H +A T D \Gamma1 A)x = b (1.1) where b is modest an...