New variable-metric algorithms for nondifferentiable optimization problems

This paper deals with new variable-metric algorithms for nonsmooth optimization problems, the so-called adaptive algorithms. The essence of these algorithms is that there are two simultaneously working gradient algorithms: the first is in the main space and the second is in the space of the matrices that modify the main variables. The convergence of these algorithms is proved for different cases. The results of numerical experiments are also given.     

Local convergence of interior-point algorithms for degenerate monotone LCP

Most asymptotic convergence analysis of interior-point algorithms for monotone linear complementarity problems assumes that the problem is nondegenerate, that is, the solution set contains a strictly complementary solution. We investigate the behavior of these algorithms when this assumption is removed.The work of this author was based on research supported by the National Science Foundation under grant DDM-9109404 and the Office of Naval Research under grant N00014-93-1-0234.The work of this author was based on research supported by the Office of Scientific Computing, U.S. Department of Energy, under Contract W-31-109-Eng-38.     

Parallel pivotal algorithm for solving the linear complementarity problem

We propose a parallel implementation of the classical Lemke's algorithm for solving the linear complementarity problem. The algorithm is designed for a loosely coupled network of computers which is characterized by relatively high communication costs. We provide an accurate prediction of speedup based on a simple operation count. The algorithm produces speedup nearp, wherep is the number of processors, when tested on large problems as demonstrated by computational results on the CRYSTAL token-ring multicomputer and the Sequent Balance 21000 multiprocessor.This material is based on research supported by National Science Foundation Grants DCR-84-20963 and DCR-850-21228 and by Air Force Office of Scientific Research Grants AFSOR-86-0172 and AFSOR-86-0255 while the author was at the University of Wisconsin, Madison, Wisconsin.     

The Cascade Algorithm for the Numerical Computation of Refinable Functions

The cascade algorithm is a method which can be used for the computation of refinable functions. We prove here that in general the cascade algorithm will not inherit the accuracy and refinability of the limit refinable function. We provide conditions for which the cascade iteration does indeed preserve accuracy.     

A barrier function method for minimax problems

This paper presents an algorithm based on barrier functions for solving semi-infinite minimax problems which arise in an engineering design setting. The algorithm bears a resemblance to some of the current interior penalty function methods used to solve constrained minimization problems. Global convergence is proven, and numerical results are reported which show that the algorithm is exceptionally robust, and that its performance is comparable, while its structure is simpler than that of current first-order minimax algorithms.This research was supported by the National Science Foundation grant ECS-8517362, the Air Force Office Scientific Research grant 86-0116, the California State MICRO program, and the United Kingdom Science and Engineering Research Council.     

关于非线性规划的同伦算法与罚函数法

本文揭示了关于非线性规划问题的同伦算法与外点罚函数法的关系，并讨论了有关同伦算法的收敛条件，给出了一些典型的检验问题的计算结果以表明利用结构的分段线性同伦算法的有效性。     

An O(n 3 L) adaptive path following algorithm for a linear complementarity problem

In this paper we propose an O(n ³ L) algorithm which is a modification of the path following algorithm [8] for a linear complementarity problem. The path following algorithm has to take a short step size in each iteration in order to bound the number of overall arithmetic operations by O(n ³ L). In practical computation, we can determine the step size adaptively. Mizuno, Yoshise, and Kikuchi [11] reported that such an adaptive algorithm required about O(L) iterations for some test problems. Here we show that we can use a rank one update technique in the adaptive algorithm so that the number of overall arithmetic operations is theoretically bounded by O(n ³ L).Research supported in part by the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell University.Research supported in part by NSF grants ECS-8602534 and DMS-8904406 and ONR contract N-00014-87-K0212.     

A penalty optimization technique for a class of regulator problems,part 3

The frequent encounter in economics, disease epidemics, resource depletion, as well as electromechanical systems of phenomena that cannot be readily modelled unless equations involving time delays are admitted has drawn a great deal of research attention to differentialdelay systems, which have consequently grown at an unprecedented rate. In Refs. 1 and 2, we provided and generalized a function space algorithm, known as the extended conjugate-gradient method (ECGM) algorithm, as a penalty optimization technique for solving a continuous optimal control problem governed by a system of differential-delay equations under the influence of certain inhomogeneous forcing terms; however, numerical results to support our analysis were not presented. To fill the gap, this paper presents the numerical behavior of the ECGM algorithm through the solution of a significant example. There is a noticeable improvement in the accuracy of the solutions obtained via the ECGM algorithm over those obtained via the averaging approximation principles developed by Banks and Burns (Ref. 3). Furthermore, the paper establishes some functional inequalities motivated from practical situations.It is a sincere pleasure to acknowledge the financial support of the University of Ilorin, Ilorin, Nigeria, Senate Research Grant No. UI/REG/SEO/57-1987.The author would like to gratefully thank Dr. J. Schiess of NASA-LRC Analysis and Computation Division and two anonymous referees for their helpful comments which led to a major improvement in the quality of the paper and especially for suggesting the inclusion of Ref. 3.     

Magnetohydrodynamic steady flow computations in three dimensions

A complete three-dimensional mathematical model has been developed governing the steady, laminar flow of an incompressible fluid subjected to a magnetic field and including internal heating due to the Joule effect, heat transfer due to conduction, and thermally induced buoyancy forces. The thermally induced buoyancy was accounted for via the Boussinesq approximation. The entire system of eight partial differential equations was solved by integrating intermittently a system of five fluid flow equations and a system of three magnetic field equations and transferring the information through source-like terms. An explicit Runge-Kutta time-stepping algorithm and a finite difference scheme with artificial compressibility were used in the general non-orthogonal curvilinear boundary-conforming co-ordinate system. Comparison of computational results and known analytical solutions in two and three dimensions demonstrates high accuracy and smooth monotone convergence of the iterative algorithm. Results of test cases with thermally induced buoyancy demonstrate the stabilizing effect of the magnetic field on the recirculating flows.     

High order starting iterates for implicit Runge-Kutta methods: an improvement for variable-step symplectic integrators

When dealing with implicit Runge–Kutta methods, the equationsdefining the stages are usually solved by iterative methods.The closer the first iterate is to the solution, the fewer iterationsare required. In this paper the author presents and analysesnew high order algorithms to compute such initial iterates.Numerical experiments are given to illustrate the performanceof the new procedures when combined with a variable-step symplecticintegrator.