A generalization of Polyak's convergence result for subgradient optimization

This paper generalizes a practical convergence result first presented by Polyak. This new result presents a theoretical justification for the step size which has been successfully used in several specialized algorithms which incorporate the subgradient optimization approach.     

On von Neumann's problem in extension theory of nonnegative operators

The solution of von Neumann's problem about parametrization of all nonegative selfadjoint extensions of a nonnegative densely defined operator in terms of his formulas is obtained.

     

Semisimplicity and von Neumann's regularity

The purpose of this paper is to link semigroup theory and category theory by using Green's relations to study regular morphisms and semisimple objects in a category which is filtering in the sense that every morphism admits an epic-monic factorization. For example, the following results are shown for a filtering category C ∶ (1)C is regular if and only if every monic inC is left invertible and every epic inC is right invertible: (2) the monoid of endomorphisms of any semisimple object ofC is regular. Communicated by A.H. Clifford     

A compact optimization model for the tail assignment problem

This paper investigates a new model for the so-called Tail Assignment Problem, which consists in assigning a well-identified airplane to each flight leg of a given flight schedule, in order to minimize total cost (cost of operating the flights and possible maintenance costs) while complying with a number of operational constraints. The mathematical programming formulation proposed is compact (i.e., involves a number of $0?1$ decision variables and constraints polynomial in the problem size parameters) and is shown to be of significantly reduced dimension as compared with previously known compact models. Computational experiments on series of realistic problem instances (obtained by random sampling from real-world data set) are reported. It is shown that with the proposed model, current state-of-the art MIP solvers can efficiently solve to exact optimality large instances representing 30-day flight schedules with typically up to 40 airplanes and 1500 flight legs connecting as many as 21 airports. The model also includes the main existing types of maintenance constraints, and extensive computational experiments are reported on problem instances of size typical of practical applications.     

A generalization of Everett's result on the Collatz 3x + 1 problem

It is a well-known conjecture that given m ε N, the set of natural numbers, the sequence {m_n}^∞_n−0, defined by the iterative formula m₀ = m,

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has some iterate m_j = 1. It is shown in this paper that for any k ε N, “almost every” natural number m greater than unity has k iterates less than m.     

A generalization of a result of Kazhdan and Lusztig

Kazhdan and Lusztig showed that every topologically nilpotent, regular semisimple orbit in the Lie algebra of a simple, split group over the field is, in some sense, close to a regular nilpotent orbit. We generalize this result to a setting that includes most quasisplit -adic groups.

     

A generalization of a result of Dembowski and Wagner

The famous Dembowski-Wagner theorem gives various characterizations of the classical geometric 2-design PG _n-1(n, q) among all 2-designs with the same parameters. One of the characterizations requires that all lines have size q + 1. It was conjectured [2] that this is also true for the designs PG _d (n, q) with 2 ≤ d ≤  n − 1. We establish this conjecture, hereby improving various previous results.     

A generalization of a result of Bauer and Schmeichel

We prove that if G is a 1-tough graph withn ≥ 3 vertices such thatd(u) + d(v) +d(w) ≥n+ κ —2 holds for any triple of independent verticesu, v andw ofG, thenG is hamiltonian, wherek is the vertex connectivity ofG. This generalizes a recent result of Baur and Schmeichel.     

A matroid generalization of a result of Dirac

This paper generalizes a theorem of Dirac for graphs by proving that ifM is a 3-connected matroid, then, for all pairs {a,b} of distinct elements ofM and all cocircuitsC ^* ofM, there is a circuit that contains {a,b} and meetsC ^*. It is also shown that, although the converse of this result fails, the specified condition can be used to characterize 3-connected matroids.The author's research was partially supported by a grant from the National Security Agency.     

A personnel assignment problem

The following personnel assignment problem is considered. Let (T, ?) be a linearly ordered set where T is a set (of people), and let (P, ?) be a partially ordered set where P, a set of positions of two types, is of the same cardinality as T. Each person i in T is to be assigned to a position. A feasible assignment of personnel to positions is an embedding of (P, ?) in (T, ?). Given measures of each person's effectiveness in both types of positions, an optimal assignment maximizes the total measure of effectiveness. The general assignment problem is shown to be NP-complete. O(n log n) algorithms for two special cases of the problem are presented.     

A generalization of the Oberwolfach problem

Let n > 1 be an integer and let a₂,a₃,…,a_n be nonnegative integers such that . Then can be factored into ‐factors, ‐factors,…, ‐factors, plus a 1‐factor. © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 151–161, 2002     

A generalized bottleneck assignment problem

A solution procedure is presented for a generalization of the standard bottleneck assignment problem in which a secondary criterion is automatically provided. A partitioning problem is modeled by this bottleneck problem to provide an example of its application.     

A multi-period machine assignment problem

In this paper, a multi-period assignment problem is studied that arises as part of a weekly planning problem at mail processing and distribution centers. These facilities contain a wide variety of automation equipment that is used to cancel, sort, and sequence the mail. The input to the problem is an equipment schedule that indicates the number of machines required for each job or operation during the day. This result is then post-processed by solving a multi-period assignment problem to determine the sequence of operations for each machine. Two criteria are used for this purpose. The first is to minimize the number of startups, and the second is to minimize the number of machines used per operation.The problem is modeled as a 0–1 integer program that can be solved in polynomial time when only the first criterion is considered. To find solutions in general, a two-stage heuristic is developed that always obtains the minimum number of startups, but not necessarily the minimum number of machines per operation. In a comparative study, high quality solutions were routinely provided by the heuristic in negligible time when compared to a commercial branch-and-bound code (Xpress). For most hard instances, the branch-and-bound code was not able to even find continuous solutions within acceptable time limits.     

A generalization of a classical zero-sum problem

In this note, we supply the details of the proof of the fact that if a₁,…,a_n+Ω(n) are integers, then there exists a subset M⊂{1,…,n+Ω(n)} of cardinality n such that the equation

     

A robust robust optimization result

We study the loss in objective value when an inaccurate objective is optimized instead of the true one, and show that “on average” this loss is very small, for an arbitrary compact feasible region.