Algorithm for finding a solution to the inclusion 0∈F(x)

Using the steepest-descent method combined with the Armijo stepsize rule, we give an algorithm for finding a solution to the inclusion 0F(x), whereF is a set-valued map with smooth support function. As an example, we consider the special caseF(x)=g(x)+K, withK being a convex cone andg a single-valued function. The relation between the present algorithm and that given by Burke and Han is also discussed.The valuable comments and helpful suggestions of the referee are gratefully acknowledged. Sincere thanks are due to Dr. J. Burke for submitting the necessary material and to Dr. C. Lemarechal for advices and encouragement.     

Unified steerable phase I-phase II method of feasible directions for semi-infinite optimization

In this paper, we complete a cycle in the construction of methods of feasible directions for solving semi-infinite constrained optimization problems. Earlier phase I-phase II methods of feasible directions used one search direction rule in all of ⁿ with two stepsize rules, one for feasible points and one for infeasible points. The algorithm presented in this paper uses both a single search direction rule and a single stepsize rule in all of ⁿ . In addition, the new algorithm incorporates a steering parameter which can be used to control the speed with which feasibility is achieved. The new algorithm is simpler to analyze and performs somewhat better than existing, first order, phase I-phase II methods. The new algorithm is globally convergent, with linear rate.The research reported herein was sponsored in part by the National Science Foundation Grant ECS-8713334, the Air Force Office of Scientific Research Contract AFOSR-86-0116, and the State of California MICRO Program Grant 532410-19900.The authors would like to thank Dr. J. Higgins for providing the C-code of Algorithm 3.1.     

A generalization of Jung's theorem

Jung's theorem establishes a relation between circumradius and diameter of a convex body. Half of the diameter can be interpreted as the maximum of circumradii of all 1-dimensional sections or 1-dimensional orthogonal projections of a convex body. This point of view leads to two series of j-dimensional circumradii, defined via sections or projections. In this paper we study some relations between these circumradii and by this we find a natural generalization of Jung's theorem.I would like to thank Prof. Dr J. M. Wills, who called my attention to these generalized circumradii.     

A Newton-type method for positive-semidefinite linear complementarity problems

The paper presents a damped and perturbed Newton-type method for solving linear complementarity problems with positive-semidefinite matricesM. In particular, the following properties hold: all occurring subproblems are linear equations; each subproblem is uniquely solvable without any assumption; every accumulation point generated by the method solves the linear complementarity problem. The additional property ofM to be an R₀-matrix is sufficient, but not necessary, for the boundedness of the iterates. Provided thatM is positive definite on a certain subspace, the method converges Q-quadratically.The author would like to thank the anonymous referees and Dr. K. Schönefeld for their valuable comments and suggestions. He is also grateful to Prof. Dr. J. W. Schmidt for his continuous interest in this study.     

GleichmÄ\ig trennscharfe Tests

Summary Generalizing a result of J. Neyman [5] necessary and sufficient conditions are given that for each simple hypothesis and each size there exists a randomized test which is uniformly most powerful against all alternatives. The class of all these tests is described explicitely.

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Zum Schlu\ möchte ich den Herren Professoren Dr. D. Bierlein, Dr. J. Pfanzagl und Dr. H. Richter für wertvolle Anregungen und Diskussionen danken.     

On methods for generating uniform random points on the surface of a sphere

Some methods for generating random points uniformly distributed on the surface of ann-sphere have been proposed to simulate spherical processes on computer. A standard method is to normalize random points inside of the sphere, see M. Muller [5]. Improved methods were given by J. M. Cook [1] and G. Marsaglia [4] in three and four dimensions, and computational methods in higher dimensions by J. S. Hicks and R. F. Wheeling [3] and M. Sibuya [6]. In this paper we shall offer direct methods for generating uniform random points on the surface of a unitn-sphere, which can be easily combined with Marsaglia's idea for getting more improved methods. Our method in even dimensions was obtained by M. Sibuya [6], but a differential-geometric view-point will make analyses simpler, even in odd dimensions.     

ODE versus SQP methods for constrained optimization

In this paper, we review some methods which are designed to solve equality constrained minimization problems by following the trajectory defined by a system of ordinary differential equations. The numerical performance of a number of these methods is compared with that of some popular sequential quadratic programming algorithms. On a set of eighteen difficult test problems, we observe that several of the ODE methods are more successful than any of the SQP techniques. We suggest that these experimental results indicate the need for research both to analyze and develop new ODE techniques and also to strengthen the currently available SQP algorithms.This work was supported by a SERC Research Studentship for the first author. Both authors are indebted to Dr. J. J. McKeown and Dr. K. D. Patel of SCICON Ltd., the collaborating establishment, for their advice and encouragement.     

Über die Bahnaffinnormalen bei der Bewegung eines starren ebenen Systems bei kollokaler Lage der ersten drei Pole

Ohne ZusammenfassungMit 6 AbbildungenHerrn Professor Dr. J. Krames zum 70. Geburtstag gewidmetProf.R. Bereis wurde am 12. 2. 1903 in Wien geboren und ist am 6. 6. 1966 in Berlin gestorben. Er verfaßte die Arbeit im Krankenhaus und widmete sie seinem verehrten Lehrer und KollegenJ. Krames.     

Book Vignettes

Shaver, Michael and Adey, Philip, Towards a Science of Science Teaching. Vannint, Vanio and Pogliana, Guiliano (eds.) and translated and revised by Jolly, Dr. Richard T. Driscoll, Mark J. Research Within Reach: Elementary School Mathematics. Frazer, Frank, Discovering Energy.     

Quasivarieties of distributivep-algebras

A counterexample is given to show that not all quasivarieties of distributivep-algebra lie between two consecutive varieties. It is shown that there are no strict relatively congruence distributive quasivarieties of distributivep-algebras. The relative congruence extension property in the class of distributivep-algebras is studied. It is proved that only quasivarieties which are varieties possess this property.Presented by J. Berman.This research is part of the author's dissertation done at Iowa State University under the supervision of Dr. Clifford Bergman.     

Two-stage stochastic hierarchical multiple risk problems: models and algorithms

In this paper, we consider a class of two-stage stochastic risk management problems, which may be stated as follows. A decision-maker determines a set of binary first-stage decisions, after which a random event from a finite set of possible outcomes is realized. Depending on the realization of this outcome, a set of continuous second-stage decisions must then be made that attempt to minimize some risk function. We consider a hierarchy of multiple risk levels along with associated penalties for each possible scenario. The overall objective function thus depends on the cost of the first-stage decisions, plus the expected second-stage risk penalties. We develop a mixed-integer 0–1 programming model and adopt an automatic convexification procedure using the reformulation–linearization technique to recast the problem into a form that is amenable to applying Benders’ partitioning approach. As a principal computational expedient, we show how the reformulated higher-dimensional Benders’ subproblems can be efficiently solved via certain reduced-sized linear programs in the original variable space. In addition, we explore several key ingredients in our proposed procedure to enhance the tightness of the prescribed Benders’ cuts and the efficiency with which they are generated. Finally, we demonstrate the computational efficacy of our approaches on a set of realistic test problems. Dr. H. D. Sherali acknowledges the support of the National Science Foundation under Grant No. DMI-0552676. Dr. J. C. Smith acknowledges the support of the Air Force Office of Scientific Research under Grant No. AFOSR/MURI F49620-03-1-0477.     

Strong convergence theorems by hybrid method for asymptotically -strict pseudo-contractive mapping in Hilbert space

In this paper, we introduce the hybrid method of modified Mann’s iteration for an asymptotically k-strict pseudo-contractive mapping. Then we prove that such a sequence converges strongly to P_F(T)x₀. This main theorem improves the result of Issara Inchan [I. Inchan, Strong convergence theorems of modified Mann iteration methods for asymptotically nonexpansive mappings in Hilbert spaces, Int. J. Math. Anal. 2 (23) (2008) 1135–1145] and concerns the result of Takahashi et al. [W. Takahashi, Y. Takeuchi, R. Kubota, Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert space, J. Math. Anal. Appl. 341 (2008) 276–286], and many others.     

Properties of the conjugate-gradient and Davidon methods

Two quadratically convergent gradient methods for minimizing an unconstrained function of several variables are examined. The heart of the Fletcher and Powell reformulation of Davidon's method is a variableH-matrix. The eigenvalues and eigenvectors of this matrix for a quadratic function are explored, leading to a proof that the gradient vectors at each step are mutually orthogonal. From this, a geometric interpretation of theH-matrix in terms of the projection of the gradient into a solution subspace is derived. These properties are then used to arrive at the main result, which states that, for a quadratic function, the direction vectors generated by the Davidon algorithm and the conjugate-gradient algorithm of Hestenes and Stiefel are scalar multiples of each other, provided the initial step each takes is in the direction of steepest descent. If one assumes no round-off error and a perfect one-dimensional search, the methods generate identical steps leading to the minimum.It is also shown that, for a quadratic function, the Davidon algorithm has a simplified version which searches in the same directions. However, the unique advantage of the original scheme, that it yields the curvature of the function at the minimum, is sacrificed for simplicity.Although these results apply to the case of a quadratic function, a comparative study of the same algorithms for a general function can be found in a companion paper.This research was carried out under Contract No. NAS 9-4036, NASA-Manned Spacecraft Center, Houston, Texas. The author is indebted to Dr. H. J. Kelley, whose suggestions and encouragement provided the inspiration for this paper.     

Consistent approximations in Newton-type decomposition methods

Summary A class of Newton-type decomposition methods for the solution of large systems of nonlinear equations recently introduced by the first two authors is extended in such a way that consistent approximations to the derivatives by appropriate difference quotients are permitted. Concrete realizations with essential merits for practical computation and a rigorous convergence analysis are given. For special choices of the discretization parameterR-orders greater than one are derived.Dedicated to Prof. Dr. Dr. h.c. Lothar Collatz on the occasion of his 75th birthday     

The Paneitz Curvature Problem on Lower-Dimensional Spheres

In this paper we prescribe a fourth order conformal invariant (the Paneitz curvature) on the n-spheres, with n∊{5,6}. Using dynamical and topological methods involving the study of critical points at infinity of the associated variational problem, we prove some existence results. Mathematics Subject Classifications (2000): 35J60, 53C21, 58J05, 35J30.     

On perfectly homogeneous bases in Banach spaces

It is proved that a Banach space is isomorphic toc _o or tol _p if and only if it has a normalized basis {χ_i i _{} i=1} ^∞ which is equivalent to every normalized block-basis with respect to {χ_i i _{} i=1} ^∞ . This is part of the author’s Ph.D. thesis prepared at the Hebrew University of Jerusalem under the supervision of Prof. A. Dvoretzky and Dr. J. Lindenstrauss. The author wishes to thank Dr. Lindenstrauss for his helpful guidance and for the interest he showed in the paper, and the referee for his valuable remakrs.     

On the sectional geometric genus of quasi-polarized varieties,II

Let (X,L) be a quasi-polarized variety of dim X=n. In the previous paper we gave a new invariant (the i-th sectional geometric genus) of (X,L), which is a generalization of the degree and the sectional genus of (X,L). In this paper we study some properties of the sectional geometric genus, and we consider the i-th sectional geometric genus of some special varieties. This research was partially supported by Grant-in-Aid for Scientific Research (No.14740018) from the Ministry of Education, Culture, Sports, Science and Technology.Mathematics Subject Classification (2000):Primary 14C20; Secondary 14J30, 14J32, 14J35, 14J40, 14J45, 14N99     

A handy proof of Gay's theorem

In 1979, Gay proved that Broyden's methods, when used for n‐square linear systems, terminate in at most 2n iterations (SIAM J. Numer. Anal. 1979; 16 :623–630). Also, the ABS methods were introduced in 1984 (Numer. Math. 1984; 45 :1361–1376). In this paper we show another (handy) proof of Gay's theorem by these algorithms. Copyright © 2008 John Wiley & Sons, Ltd.     

Eine Bemerkung zum Satz von Ivory

In 1809 J. IVORY found a distance relation between any two confocal ellipsoids in Euclidean space E³. After that, the theorem of IVORY has been formulated for any two nondegenerate confocal quadrics of the same type. In this paper we succeed in completing these results and in formulating and proving them uniformly for any two quadrics of the same type belonging to a general system of confocal quadrics.

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Herrn Prof. Dr. Oswald Giering zum 60. Geburtstag gewidmet     

Einige Schranken für die Absolutbeträge der Wurzeln algebraischer Gleichungen

Ohne ZusammenfassungMeinem verehrten Lehrer, Herrn Prof. Dr.J. Heinhold, T. H. München, möchte ich an dieser Stelle für wertvolle Anregungen meinen ergebensten Dank aussprechen.