The Newton modified barrier method for QP problems

The Modified Barrier Functions (MBF) have elements of both Classical Lagrangians (CL) and Classical Barrier Functions (CBF). The MBF methods find an unconstrained minimizer of some smooth barrier function in primal space and then update the Lagrange multipliers, while the barrier parameter either remains fixed or can be updated at each step. The numerical realization of the MBF method leads to the Newton MBF method, where the primal minimizer is found by using Newton's method. This minimizer is then used to update the Lagrange multipliers. In this paper, we examine the Newton MBF method for the Quadratic Programming (QP) problem. It will be shown that under standard second-order optimality conditions, there is a ball around the primal solution and a cut cone in the dual space such that for a set of Lagrange multipliers in this cut cone, the method converges quadratically to the primal minimizer from any point in the aforementioned ball, and continues, to do so after each Lagrange multiplier update. The Lagrange multipliers remain within the cut cone and converge linearly to their optimal values. Any point in this ball will be called a hot start. Starting at such a hot start, at mostO(In In ^-1) Newton steps are sufficient to perform the primal minimization which is necessary for the Lagrange multiplier update. Here, >0 is the desired accuracy. Because of the linear convergence of the Lagrange multipliers, this means that onlyO(In ^-1)O(In In ^-1) Newton steps are required to reach an -approximation to the solution from any hot start. In order to reach the hot start, one has to perform Newton steps, wherem characterizes the size of the problem andC>0 is the condition number of the QP problem. This condition number will be characterized explicitly in terms of key parameters of the QP problem, which in turn depend on the input data and the size of the problem.Partially supported by NASA Grant NAG3-1397 and National Science Foundation Grant DMS-9403218.     

A Note on π-Engel Conditions

In this paper we show that the weakly -Engel conditions are closely related to the existance of normal -complements; while the -Engel conditions are closely related to the -nilpotent groups.AMS Subject Classification (2000): 20D20     

ɛ-duality theorem of nondifferentiable nonconvex multiobjective programming

Necessary Kuhn-Tucker conditions up to precision without constraint qualification for -Pareto optimality of multiobjective programming are derived. This article suggests the establishment of a Wolfe-type -duality theorem for nondifferentiable, nonconvex, multiobjective minimization problems. The -vector Lagrangian and the generalized -saddle point for Pareto optimality are studied.     

μ-Statistically Convergent Function Sequences

In the present paper we are concerned with convergence in -density and -statistical convergence of sequences of functions defined on a subset D of real numbers, where is a finitely additive measure. Particularly, we introduce the concepts of -statistical uniform convergence and -statistical pointwise convergence, and observe that -statistical uniform convergence inherits the basic properties of uniform convergence.     

On the Operator Semirings of a Γ-Semiring

In this paper we introduce the notion of operator semirings of a -semiring to study -semirings. It is shown that the lattices of all left (right) ideals (two-sided ideals) of a -semiring and its right (respectively left) operator semiring are isomorphic. This has many applications to characterize various -semirings.AMS Subject Classification (2000): 16Y60, 16Y99     

Metric and Complete Metric σ-Frames

A -frame is a lattice in which countable joins exist and binary meets distribute over countable joins. In this paper, the category MFrm, of metric -frames, is introduced, and it is shown to be equivalent to the category MLFrm _u, of metric Lindelöf frames.Finally, it is shown that the complete metric -frames are exactly the cozero parts of complete metric Lindelöf frames.     

Existence of ∈-Minima for Vector Optimization Problems

The -generalized minima for vector optimization problems are defined and a sufficient condition for the existence of -generalized minima for vector optimization problems is established.     

(φ,ξ,η,λ)‐Structures of Parabolic Type

We consider (,,,)structures of parabolic type on hypersurfaces of dual spaces and study the rank of the affinor . We consider almost contact metric structures of parabolic type of the first kind on hypersurfaces of 4dimensional dual metric space. We study the properties of these structures and give examples of normal, integrable, and Sasakian parabolic structures.     

Stability Tests for Two Common Classes of Linear Time‐Delay and Hybrid Systems

The stability of linear timedelay systems with point internal delays is difficult to deal with in practice because of the fact that their characteristic equation is usually of transcendent type rather than of polynomial type. This feature usually causes the system to possess an infinite number of poles. In this paper, stability tests for this class of systems are obtained based either on extensions of classical tests applicable to delayfree systems or on approaches within the framework of twodimensional digital filters. Some of those twodimensional stability tests are also proved to be useful for stability testing of a common class of linear hybrid systems which involve coupled continuous and digital substates after a slightadhoc adaptation of the tests for that situation.     

A Characterization of Markov Property for Semigroups with Invariant Measure

The positivity of the carré du champ operator is a direct consequence of the positivity of the associated Markov semigroup. We show in this note that the reciprocal implication holds in invariant measure, under minimal hypotheses of continuity and stability.     

⊕-Cofinitely Supplemented Modules

Let R be a ring and M a right R-module. M is called -cofinitely supplemented if every submodule N of M with M/N finitely generated has a supplement that is a direct summand of M. In this paper various properties of the -cofinitely supplemented modules are given. It is shown that (1) Arbitrary direct sum of -cofinitely supplemented modules is -cofinitely supplemented. (2) A ring R is semiperfect if and only if every free R-module is -cofinitely supplemented. In addition, if M has the summand sum property, then M is -cofinitely supplemented iff every maximal submodule has a supplement that is a direct summand of M.     

Von Neumann Categories and Extended L2 Cohomology

In this paper we suggest a new general formalism for studying the L2 invariants of polyhedra and manifolds. First, we examine generality in which one may apply the construction of the extended Abelian category, which was earlier suggested by the author using the ideas of P. Freyd. This leads to the notions of a finite von Neumann category and a trace on such a category. Given a finite von Neumann category, we study the extended L2 homology and cohomology theories with values in the Abelian extension. Any trace on the initial category produces numerical invariants – the von Neumann dimension and the Novikov–Shubin numbers. Thus, we obtain the local versions of the Novikov–Shubin invariants, localized at different traces. In the Abelian case this localization can be made more geometric: we show that any torsion object determines a divisor – a closed subspace of the space of the parameters. The divisors of torsion objects together with the information produced by the local Novikov–Shubin invariants may be used to study multiplicities of intersections of algebraic and analytic varieties (we discuss here only simple examples demonstrating this possibility). We compute explicitly the divisors and the von Neumann dimensions of the extended L2 cohomology in the real analytic situation. We also give general formulae for the extended L2 cohomology of a mapping torus. Finally, we show how one can define a De Rham version of the extended cohomology and prove a De Rhamtype theorem.     

A New Approach to Multiobjective Programming with a Modified Objective Function

In this paper, optimality for multiobjective programming problems having invex objective and constraint functions (with respect to the same function ) is considered. An equivalent vector programming problem is constructed by a modification of the objective function. Furthermore, an -Lagrange function is introduced for a constructed multiobjective problem and modified saddle point results are presented.     

Start‐up class models in multiple‐class queues with N‐policy

In this paper, we consider multipleclass queueing systems with Npolicy in which the idle server starts service as soon as the number of customers in the startup class reaches threshold N. We consider the cases of FCFS and nonpreemptive priority. We obtain the Laplace–Stieltjes transform of the waiting times of each class of customers. We also show some results for the general behavior of such systems.     

The η-Invariant of Twisted Dirac Operators of S3/Γ

The aim of this paper is to compute the and ~-invariants for the Dirac operator of the quotient of the sphere S ³ by a finite subgroup, twisted by a representation of its fundamental group.     

Relative E ‐Theory

The Etheory of A. Connes and N. Higson provides a new realization of Khomology based on the notion of asymptotic morphisms. In this paper we define relative Etheory, associating to a C*algebra A and an ideal I the Abelian groups Enrel(A;I). These groups are related to the Etheory groups of A and I in the familiar way by a long exact sequence and excision isomorphisms. The definition of relative Etheory is motivated by the properties of first order, elliptic differential operators on complete Riemannian manifolds. Applications will be considered in a future publication.     

Relationship between solid cones and cones with bases

This paper establishes a relation between -based cones and solid cones in a separated locally convex topological vector space. The relation reveals that the properties of the base of a based cone affect deeply the topological properties of the dual cone.The author would like to thank Professor C. C. Huang for his discussion on the proof of Corollary 2.1. The author is also grateful to the referees for helpful comments and suggestions.     

Production systems with interruptions, arbitrary topology and finite buffers

We consider a production system with finite buffers and arbitrary topology where service time is subject to interruptions in one of three ways, viz. machine breakdown, machine vacations or Npolicy. We develop a unified approximation (analytical) methodology to calculate the throughput of the system using queueing networks together with decomposition, isolation and expansion techniques. The methodology is rigorously tested covering a large experimental region. Orthogonal arrays are used to design the experiments in order to keep the number of experiments manageable. The results obtained using the approximation methodology are compared to the simulation results. The ttests carried out to investigate the differences between the two results show that they are statistically insignificant. Finally, we test the methodology by applying it to several arbitrary topology networks. The results show that the performance of the approximation methodology is consistent, robust and produces excellent results in a variety of experimental conditions.     

ɛ-Mixed strategies for static continuous-kernel stackelberg games

We consider a two-level optimization problem corresponding to a Stackelberg game in which the response function of the follower is multivalued and some convexity assumption on the data is missing. Different notions of -mixed strategies are introduced and sufficient conditions on the data for existence and convergence of the corresponding average values, as converges to 0, are given together with several examples.This paper is based upon results first presented at the International Conference on Game Theory, Firenze, Italia, 1991. This research has been supported by MPI, Roma, Italia.     

Prime Rings with Involution Equipped with Some New Product

Let R be a ring with involution *. We consider R as a ring equipped with a new product r s = rs + sr^*. The relationship between (ordinary) ideals of R and right ideals of R with respect to the product is studied.AMS Subject Classification (2000): 16W10, 16D25