A constraint linearization method for nondifferentiable convex minimization

Summary This paper presents a readily implementable algorithm for solving constrained minimization problems involving (possibly nonsmooth) convex functions. The constraints are handled as in the successive quadratic approximations methods for smooth problems. An exact penalty function is employed for stepsize selection. A scheme for automatic limitation of penalty growth is given. Global convergence of the algorithm is established, as well as finite termination for piecewise linear problems. Numerical experience is reported.Sponsored by Program CPBP 02.15     

Algorithms for bound constrained quadratic programming problems

Summary We present an algorithm which combines standard active set strategies with the gradient projection method for the solution of quadratic programming problems subject to bounds. We show, in particular, that if the quadratic is bounded below on the feasible set then termination occurs at a stationary point in a finite number of iterations. Moreover, if all stationary points are nondegenerate, termination occurs at a local minimizer. A numerical comparison of the algorithm based on the gradient projection algorithm with a standard active set strategy shows that on mildly degenerate problems the gradient projection algorithm requires considerable less iterations and time than the active set strategy. On nondegenerate problems the number of iterations typically decreases by at least a factor of 10. For strongly degenerate problems, the performance of the gradient projection algorithm deteriorates, but it still performs better than the active set method.Work supported in part by the Applied Mathematical Sciences subprogram of the Office of Energy Research of the U.S. Department of Energy under Contract W-31-109-Eng-38     

Accelerated Gauss-Newton algorithms for nonlinear least squares problems

Recent theoretical and practical investigations have shown that the Gauss-Newton algorithm is the method of choice for the numerical solution of nonlinear least squares parameter estimation problems. It is shown that when line searches are included, the Gauss-Newton algorithm behaves asymptotically like steepest descent, for a special choice of parameterization. Based on this a conjugate gradient acceleration is developed. It converges fast also for those large residual problems, where the original Gauss-Newton algorithm has a slow rate of convergence. Several numerical test examples are reported, verifying the applicability of the theory.     

Über die globale Konvergenz von Variable-Metrik-Verfahren mit nicht-exakter Schrittweitenbestimmung

Summary We consider unconstrained minimization problems and the application of the Broyden-Fletcher-Goldfarb-Shanno variable metric algorithm without exact line searches. For a certain class of step functions the global convergence of this method is proven, generalizing a result given by Powell. Furthermore some remarks are made concerning the superlinear convergence of this particular variable metric algorithm.

     

Multi-mode resource availability cost problem with recruitment and release dates for resources

This paper investigates the multi-mode resource availability cost problem with recruitment and release dates for resources. This problem is a more realistic model and extended case of the resource availability cost problem. The project contains activities interrelated by finish–start precedence relations with zero time lags, which require a set of renewable resources. First, a mixed integer programming formulation is proposed for the problem. Then, simulated annealing (SA) algorithm is proposed to obtain a satisfying solution for this NP-hard problem. The effectiveness of the proposed algorithm is demonstrated through comprehensive experimentation based on 300 test problems. The results are analyzed and discussed.     

Tchebychev acceleration technique for large scale nonsymmetric matrices

Summary The acceleration by Tchebychev iteration for solving nonsymmetric eigenvalue problems is dicussed. A simple algorithm is derived to obtain the optimal ellipse which passes through two eigenvalues in a complex plane relative to a reference complex eigenvalue. New criteria are established to identify the optimal ellipse of the eigenspectrum. The algorithm is fast, reliable and does not require a search for all possible ellipses which enclose the spectrum. The procedure is applicable to nonsymmetric linear systems as well.     

A multi-grid algorithm for mixed problems with penalty

Summary In this paper we describe a multi-grid algorithm for the finite element approximation of mixed problems with penalty by the MINI-element. It is proved that the convergence rate of the algorithm is bounded away from 1 independently of the meshsize and of the penalty parameter. For convenience, we only discuss Jacobi relaxation as smoothing operator in detail.The paper was written during the author's stay at the Ruhr-Universität Bochum and revised by D. Braess after the author's return to China     

A parallel projection method for solving generalized linear least-squares problems

Summary A parallel projection algorithm is proposed to solve the generalized linear least-squares problem: find a vector to minimize the 2-norm distance from its image under an affine mapping to a closed convex cone. In each iteration of the algorithm the problem is decomposed into several independent small problems of finding projections onto subspaces, which are simple and can be tackled parallelly. The algorithm can be viewed as a dual version of the algorithm proposed by Han and Lou [8]. For the special problem under consideration, stronger convergence results are established. The algorithm is also related to the block iterative methods of Elfving [6], Dennis and Steihaug [5], and the primal-dual method of Springarn [14].This material is based on work supported in part by the National Science foundation under Grant DMS-8602416 and by the Center for Supercomputing Research and Development, University of Illinois at Urbana-Champaign     

GLOBAL CONVERGENCE OF QPFTH METHOD FOR LARGE-SCALE NONLINEAR SPARSE CONSTRAINED OPTIMIZATION

1.IntroductionIn[6],aQPFTHmethodwasproposedforsolvingthefollowingnonlinearprogrammingproblemwherefunctionsf:R"-- RIandgi:R"-- R',jeJaretwicecontinuouslydifferentiable.TheQPFTHalgorithmwasdevelopedforsolvingsparselarge-scaleproblem(l.l)andwastwo-stepQ-quadraticallyandR-quadraticallyconvergent(see[6]).Theglobalconvergenceofthisalgorithmisdiscussedindetailinthispaper.Forthefollowinginvestigationwerequiresomenotationsandassumptions.TheLagrangianofproblem(1.1)isdefinedbyFOundationofJiangs…     

An interval branch and bound algorithm for bound constrained optimization problems

In this paper, we propose modifications to a prototypical branch and bound algorithm for nonlinear optimization so that the algorithm efficiently handles constrained problems with constant bound constraints. The modifications involve treating subregions of the boundary identically to interior regions during the branch and bound process, but using reduced gradients for the interval Newton method. The modifications also involve preconditioners for the interval Gauss-Seidel method which are optimal in the sense that their application selectively gives a coordinate bound of minimum width, a coordinate bound whose left endpoint is as large as possible, or a coordinate bound whose right endpoint is as small as possible. We give experimental results on a selection of problems with different properties.     

Analysis of a damped nonlinear multilevel method

Summary In this paper, we present a new algorithm that is obtained by introducing a damping parameter in the classical Nonlinear Multilevel Method. We analyse this Damped Nonlinear Multilevel Method. In particular, we prove global convergence and local efficiency for a suitable class of problems.     

A predictor-corrector technique for constrained least-squares regularization

Summary The paper describes a numerical strategy for the approximate solution of nonlinear, discretized, inverse problems by regularization. It is assumed that the solution of the associated direct problems and the computation of Fréchet derivatives are expensive. In order to minimize the amount of work, a predictor-corrector type algorithm is proposed. From a series of solutions to problems with a coarse discretization one obtains a starting approximation for a problem with a fine discretization.     

Approximation of optimal distributed control problems governed by variational inequalities

Summary A method for approximating the optimal control and the optimal state for a class of distributed control problems governed by variational inequalities is given. It uses a Rayleigh-Ritz-Galerkin scheme, regularising techniques and a gradient algorithm. A numerical example is given.     

A branch-and-bound algorithm for the resource-constrained project scheduling problem

We describe a time-oriented branch-and-bound algorithm for the resource-constrained project scheduling problem which explores the set of active schedules by enumerating possible activity start times. The algorithm uses constraint-propagation techniques that exploit the temporal and resource constraints of the problem in order to reduce the search space. Computational experiments with large, systematically generated benchmark test sets, ranging in size from thirty to one hundred and twenty activities per problem instance, show that the algorithm scales well and is competitive with other exact solution approaches. The computational results show that the most difficult problems occur when scarce resource supply and the structure of the resource demand cause a problem to be highly disjunctive.     

Quadratic convergence of potential-reduction methods for degenerate problems

Global and local convergence properties of a primal-dual interior-point pure potential-reduction algorithm for linear programming problems is analyzed. This algorithm is a primal-dual variant of the Iri-Imai method and uses modified Newton search directions to minimize the Tanabe-Todd-Ye (TTY) potential function. A polynomial time complexity for the method is demonstrated. Furthermore, this method is shown to have a unique accumulation point even for degenerate problems and to have Q-quadratic convergence to this point by an appropriate choice of the step-sizes. This is, to the best of our knowledge, the first superlinear convergence result on degenerate linear programs for primal-dual interior-point algorithms that do not follow the central path. Received: February 12, 1998 / Accepted: March 3, 2000?Published online January 17, 2001     

A new branching rule for the branch and bound algorithm for solving nonlinear integer programming problems

We examine a branch and bound algorithm for solving nonlinear (convex) integer programming problems. In this note we generalize previous results for the quadratic case. The variables are branched in such a way that the number of branch and bound nodes checked in the process is small. Numerical results confirm the efficiency.     

On improving the absolute stability of local extrapolation

Summary A widely used technique for improving the accuracy of solutions of initial value problems in ordinary differential equations is local extrapolation. It is well known, however, that when using methods appropriate for solving stiff systems of ODES, the stability of the method can be seriously degraded if local extrapolation is employed. This is due to the fact that performing local extrapolation on a low order method is equivalent to using a higher order formula and this high order formula may not be suitable for solving stiff systems. In the present paper a general approach is proposed whereby the correction term added on in the process of local extrapolation is in a sense a rational, rather than a polynomial, function. This approach allows high order formulae with bounded growth functions to be developed. As an example we derive anA-stable rational correction algorithm based on the trapezoidal rule. This new algorithm is found to be efficient when low accuracy is requested (say a relative accuracy of about 1%) and its performance is compared with that of the more familiar Richardson extrapolation method on a large set of stiff test problems.     

On the convergence of a new trust region algorithm

Summary. In this paper we present a new trust region algorithm for general nonlinear constrained optimization problems. The algorithm is based on the exact penalty function. Under very mild conditions, global convergence results for the algorithm are given. Local convergence properties are also studied. It is shown that the penalty parameter generated by the algorithm will be eventually not less than the norm of the Lagrange multipliers at the accumulation point. It is proved that the method is equivalent to the sequential quadratic programming method for all large , hence superlinearly convergent results of the SQP method can be applied. Numerical results are also reported. Received March 21, 1993     

On the fast solutions of nonlinear elliptic equations

Summary In a recent paper we described a multi-grid algorithm for the numerical solution of Fredholm's integral equation of the second kind. This multi-grid iteration of the second kind has important applications to elliptic boundary value problems. Here we study the treatment of nonlinear boundary value problems. The required amount of computational work is proportional to the work needed for a sequence of linear equations. No derivatives are required since these linear problems are not the linearized equations.     

A stepsize control for continuation methods and its special application to multiple shooting techniques

Summary A numerically applicable stepsize control for discrete continuation methods of orderp is derived on a theoretical basis. Both the theoretical results and the performance of the proposed algorithm are invariant under affine transformation of the nonlinear system to be solved. The efficiency and reliability of the method is demonstrated by solving three real life two-point boundary value problems using multiple shooting techniques. In two of the examples bifurcations occur and are significantly marked by sharp changes in the stepsize estimates.