Jacobi transplantation revisited

A transplantation theorem for Jacobi series proved by Muckenhoupt is reinvestigated by means of a suitable variant of Calderón–Zygmund operator theory. An essential novelty of our paper is weak type (1,1) estimate for the Jacobi transplantation operator, located in a fairly general weighted setting. Moreover, L ^p estimates are proved for a class of weights that contains the class admitted in Muckenhoupt’s theorem. Research of ó. Ciaurri and K. Stempak was supported by the grant MTM2006-13000-C03-03 of the DGI. Research of A. Nowak and K. Stempak was supported by MNiSW Grant N201 054 3214285.     

Optimization Techniques for State-Constrained Control and Obstacle Problems

The design of control laws for systems subject to complex state constraints still presents a significant challenge. This paper explores a dynamic programming approach to a specific class of such problems, that of reachability under state constraints. The problems are formulated in terms of nonstandard minmax and maxmin cost functionals, and the corresponding value functions are given in terms of Hamilton-Jacobi-Bellman (HJB) equations or variational inequalities. The solution of these relations is complicated in general; however, for linear systems, the value functions may be described also in terms of duality relations of convex analysis and minmax theory. Consequently, solution techniques specific to systems with a linear structure may be designed independently of HJB theory. These techniques are illustrated through two examples.The first author was supported by the Russian Foundation for Basic Research, Grant 03-01-00663, by the program Universities of Russia, Grant 03.03.007, and by the program of the Russian Federation President for the support of scientific research in leading scientific schools, Grant NSh-1889.2003.1.The second author was supported by the National Science and Engineering Research Council of Canada and by ONR MURI Contract 79846-23800-44-NDSAS.The third and first authors were supported by NSF Grants ECS-0099824 and ECS-0424445.Communicated by G. Leitmann     

An augmented Lagrangian technique for variational inequalities

A general framework for the treatment of a class of elliptic variational inequalities by an augmented Lagrangian method, when inequalities with infinite-dimensional image space are augmented, is developed. Applications to the obstacle problem, the elastoplastic torsion problem, and the Signorini problem are given.The research of the first author was supported in part by the Air Force Office of Scientific Research under Grants AFOSR-84-0398 and AFOSR-85-0303, by the National Aeronautics and Space Administration under Grant NAG-1-1517, and by NSF under Grant No. UINT-8521208. The second author's research was supported in part by the Fonds zur Förderung der wissenschaftlichen Forschung under S3206 and P6005.     

Sensitivity of Pareto Solutions in Multiobjective Optimization

The paper presents a sensitivity analysis of Pareto solutions on the basis of the Karush-Kuhn-Tucker (KKT) necessary conditions applied to nonlinear multiobjective programs (MOP) continuously depending on a parameter. Since the KKT conditions are of the first order, the sensitivity properties are considered in the first approximation. An analogue of the shadow prices, well known for scalar linear programs, is obtained for nonlinear MOPs. Two types of sensitivity are investigated: sensitivity in the state space (on the Pareto set) and sensitivity in the cost function space (on the balance set) for a vector cost function. The results obtained can be used in applications for sensitivity computation under small variations of parameters. Illustrative examples are presented.Research of this author was partially supported by Grant BEC2003-09067-C04-03.Research of this author was partially supported by NSERC Grant RGPIN-3492-00.Research of this author was partially supported by Grant BEC2003-09067-C04-02.     

On n-Contractive and n-Hypercontractive Operators, II

This paper studies the n-contractive and n-hypercontractive Hilbert space operators (n = 1, 2, . . .), classes weaker than, but related to, the class of subnormal operators. The k-hyponormal operators are the more thoroughly explored examples of classes weaker than subnormal; we show that k-hyponormality implies 2k-contractivity. Turning to weighted shifts, it is shown that if a weighted shift is extremal in the sense that the general nonnegativity test for n-contractivity is satisfied with equality to zero, then the shift is necessarily the unweighted unilateral shift. Also considered are the n-contractivity of back step extensions and perturbations of subnormal weighted shifts and some connections with the Berger measure of a subnormal shift. The second author was supported by the Korean Research Foundation Grant funded by the Korean Government (MOEHRD) (KRF-2006-312-C00027). The third author was supported by the Korean Research Foundation Grant funded by the Korean Government (MOEHRD) (KRF- 2007-359-C00005.     

Transplantation and multiplier theorems for Fourier-Bessel expansions

Proved are weighted transplantation inequalities for Fourier-Bessel expansions. These extend known results on this subject by considering the largest possible range of parameters, allowing more weights and admitting a shift. The results are then used to produce a fairly general multiplier theorem with power weights for considered expansions. Also fractional integral results and conjugate function norm inequalities for these expansions are proved.

     

Euclidean semi-matchings of random samples

A linear programming relaxation of the minimal matching problem is studied for graphs with edge weights determined by the distances between points in a Euclidean space. The relaxed problem has a simple geometric interpretation that suggests the name minimal semi-matching. The main result is the determination of the asymptotic behavior of the length of the minimal semi-matching. It is analogous to the theorem of Beardwood, Halton and Hammersley (1959) on the asymptotic behavior of the traveling salesman problem. Associated results on the length of non-random Euclidean semi-matchings and large deviation inequalities for random semi-matchings are also given.Research supported in part by NSF Grant #DMS-8812868, ARO contract DAAL03-89-G-0092.P001, AFOSR-89-08301.A and NSA-MDA-904-89-2034.     

Best Approximation and Perturbation Property in Hilbert Spaces

We study the perturbation property of best approximation to a set defined by an abstract nonlinear constraint system. We show that, at a normal point, the perturbation property of best approximation is equivalent to an equality expressed in terms of normal cones. This equality is related to the strong conical hull intersection property. Our results generalize many known results in the literature on perturbation property of best approximation established for a set defined by a finite system of linear/nonlinear inequalities. The connection to minimization problem is considered.The authors thank the referees for valuable suggestions.K.F. Ng - This author was partially supported by Grant A0324638 from the National Natural Science Foundation of China and Grants (2001) 01GY051-66 and SZD0406 from Sichuan Province. Y.R. He -This author was supported by a Direct Grant (CUHK) and an Earmarked Grant from the Research Grant Council of Hong Kong.     

Proximal Point Algorithms for General Variational Inequalities

This paper presents a unified framework of proximal point algorithms (PPAs) for solving general variational inequalities (GVIs). Some existing PPAs for classical variational inequalities, including both the exact and inexact versions, are extended to solving GVIs. Consequently, several new PPA-based algorithms are proposed. M. Li was supported by NSFC Grant 10571083 and SRFDP Grant 200802861031. L.Z. Liao was supported in part by grants from Hong Kong Baptist University and the Research Grant Council of Hong Kong. X.M. Yuan was supported in part by FRG/08-09/II-40 from Hong Kong Baptist University and NSFC Grant 10701055.     

Singular Measures and Convolution Operators

We show that in the study of certain convolution operators, functions can be replaced by measures without changing the size of the constants appearing in weak type （1, 1） inequalities. As an application, we prove that the best constants for the centered Hardy-Littlewood maximal operator associated with parallelotopes do not decrease with the dimension.     

Blow-up Results for Some Nonlinear Delay Differential Equations

In this work we study the blow up phenomena for some scalar delay differential equations. In particular, we make connection with the blow up of ordinary differential equations that are related to the delay differential equations. The first author is supported by a Grant from TWAS under contract No: 03-030 RG/MATHS/AF/AC. The second author is supported by a grant from the Lebanese National Council for Scientific Research.     

Korn’s Inequalities for Junctions of Elastic Bodies with Thin Plates

We prove the anisotropic Korn inequalities for an elastic junction of a massive body with thin plates clamped along parts of the lateral surfaces. The distribution of the weight factors in the norms under consideration depends essentially on the disposition of the plates, the way they are clamped to the body, and their relative rigidity (as compared with the body).Original Russian Text Copyright © 2005 Nazarov S. A.The author was supported by the Russian Foundation for Basic Research (Grant 03-01-00835).__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 876–889, July–August, 2005.     

Superlinear and quadratic convergence of primal-dual interior-point methods for linear programming revisited

Recently, Zhang, Tapia, and Dennis (Ref. 1) produced a superlinear and quadratic convergence theory for the duality gap sequence in primal-dual interior-point methods for linear programming. In this theory, a basic assumption for superlinear convergence is the convergence of the iteration sequence; and a basic assumption for quadratic convergence is nondegeneracy. Several recent research projects have either used or built on this theory under one or both of the above-mentioned assumptions. In this paper, we remove both assumptions from the Zhang-Tapia-Dennis theory.Dedicated to the Memory of Magnus R. Hestenes, 1906–1991This research was supported in part by NSF Cooperative Agreement CCR-88-09615 and was initiated while the first author was at Rice University as a Visiting Member of the Center for Research in Parallel Computation.The authors thank Yinyu Ye for constructive comments and discussions concerning this material.This author was supported in part by NSF Grant DMS-91-02761 and DOE Grant DE-FG05-91-ER25100.This author was supported in part by AFOSR Grant 89-0363, DOE Grant DE-FG05-86-ER25017, and ARO Grant 9DAAL03-90-G-0093.     

Best simultaneous approximation to totally bounded sequences in Banach spaces

This paper is concerned with the problem of best weighted simultaneous approximations to totally bounded sequences in Banach spaces. Characterization results from convex sets in Banach spaces are established under the assumption that the Banach space is uniformly smooth. The first author is supported in part by Scientific Research Fund of Hunan Provincial Education Department (Grant No. 06C651); the second author is supported in part by the National Natural Science Foundation of China (Grant Nos. 10671175, 10731060) and Program for New Century Excellent Talents in University; the third author is supported in part by Projects MTM2006-13997-C02-01 and FQM-127 of Spain     

Nonmonotone curvilinear line search methods for unconstrained optimization

We present a new algorithmic framework for solving unconstrained minimization problems that incorporates a curvilinear linesearch. The search direction used in our framework is a combination of an approximate Newton direction and a direction of negative curvature. Global convergence to a stationary point where the Hessian matrix is positive semidefinite is exhibited for this class of algorithms by means of a nonmonotone stabilization strategy. An implementation using the Bunch-Parlett decomposition is shown to outperform several other techniques on a large class of test problems.The work of this author was based on research supported by the National Science Foundation Grant CCR-9157632, the Air Force Office of Scientific Research Grant F49620-94-1-0036 and the Department of Energy Grant DE-FG03-94ER61915.These authors were partially supported by Agenzia Spaziale Italiana, Roma, Italy.     

A second-order method for the general nonlinear programming problem

This paper presents a multiplier-type method for nonlinear programming problems with both equality and inequality constraints. Slack variables are used for the inequalities. The penalty coefficient is adjusted automatically, and the method converges quadratically to points satisfying second-order conditions.The work of the first author was supported by NSF RANN and JSEP Contract No. F44620-71-C-0087; the work of the second author was supported by the National Science Foundation Grant No. ENG73-08214A01 and US Army Research Office Durham Contract No. DAHC04-73-C-0025.     

Control of a hybrid conditionally linear Gaussian process

A control problem is considered where the coefficients of the linear dynamics are functions of a noisily observed Markov chain. The approximation introduced is to consider these coefficients as functions of the filtered estimate of the state of the chain; this gives rise to a finite-dimensional conditional Kalman filter. A minimum principle and a new equation for an adjoint process are obtained.This research was partially supported by NSERC under Grant A-7964, by the US Air Force Office of Scientific Research under Contract AFOSR-86-0332, and by the US Army Research Office under Contract DAAL03-87-0102.The authors obtained these results during a visit to UCSD by the first author in January 1990. This author wishes to thank Professor D. D. Sworder and his department for their hospitality.     

Modified predictor-corrector algorithm for locating weighted centers in linear programming

In certain applications of linear programming, the determination of a particular solution, the weighted center of the solution set, is often desired, giving rise to the need for algorithms capable of locating such center. In this paper, we modify the Mizuno-Todd-Ye predictor-corrector algorithm so that the modified algorithm is guaranteed to converge to the weighted center for given weights. The key idea is to ensure that iterates remain in a sequence of shrinking neighborhoods of the weighted central path. The modified algorithm also possesses polynomiality and superlinear convergence.The work of the first author was supported in part by NSF Grant DMS-91-02761 and DOE Contract DE-FG05-91-ER25100.The work of the second author was supported in part by NSF Cooperative Agreement CCR-88-09615.     

Multiperiod Mean-Variance Optimization with Intertemporal Restrictions

We investigate mean-variance optimization problems that arise in portfolio selection. Restrictions on intermediate expected values or variances of the portfolio are considered. Some explicit procedures for obtaining the solution are presented. The main advantage of this technique is that it is possible to control the intermediate behavior of a portfolio’s return or variance. Some examples illustrating these situations are presented. The first author received financial support from CNPq (Brazilian National Research Council) Grants 472920/03-0 and 304866/03-2, FAPESP (Research Council of the State of S?o Paulo) Grant 03/06736-7, PRONEX Grant 015/98, and IM-AGIMB.     

Solving nonlinear inequalities in a finite number of iterations

This paper describes a modified Newton algorithm for solving a finite system of inequalities in a finite number of iterations.This research was supported by NSF Grant No. ENG-73-08214-AO1, by NSF-RANN Grant No. ENV-76-04264, and by the United Kingdom Science Research Council.