Rational solutions of the graphsack problem

A graphsack problem is a certain binary linear optimization problem with applications in optimal network design. From there a rational graphsack problem is derived by allowing the variables to vary continuously between 0 and 1. In this paper we deal with rational graphsack problems. First we develop the concept of compressed solutions and the concept of augmenting cuts. Making use of these concepts a very simple optimality criterion is derived. Finally an efficient algorithm solving rational graphsack problems is given which is polynomially bounded in time and which is closely related to the simplex algorithm.     

On cone orderings and the linear complementarity problem

This paper first generalizes a characterization of polyhedral sets having least elements, which is obtained by Cottle and Veinott [6], to the situation in which Euclidean space is partially ordered by some general cone ordering (rather than the usual ordering). We then use this generalization to establish the following characterization of the class $\text{C}$ of matrices ($\text{C}$ arises as a generalization of the class of Z-matrices; see [4], [13], [14]): M∈$\text{C}$ if and only if for every vector q for which the linear complementarity problem (q,M) is feasible, the problem (q,M) has a solution which is the least element of the feasible set of (q,M) with respect to a cone ordering induced by some simplicial cone. This latter result generalizes the characterizations of K-and Z-matrices obtained by Cottle and Veinott [6] and Tamir [21], respectively.     

Minimum information estimation of allocation models of different sizes

For allocation models consisting of n = 10, 6 and 4 equations, constrained generalized least squares coefficient estimates are compared with those obtained from the minimum information (MI) criterion in the sense of statistical information theory. The MI estimates are more efficient for n = 10. The bootstrap procedure is illustrated for the assessment of the variability of these estimates.     

A note on bilinear time series models

Conditions for the existence of a stationary solution for certain forms of bilinear difference equations are derived.     

On the hellinger square integral with respect to an operator valued measure and stationary processes

A construction of the Hellinger square integral with respect to a semispectral measure in a Banach space B is given. It is proved that the space of values of a B-valued stationary stochastic process is unitarily isomorphic to the space of all B¹-valued measures that are Hellinger square integrable with respect to the spectral measure of the process. Some applications of the above theorem in the prediction theory (especially to interpolation problem) are also considered.     

关于线性回归模型的有偏估计

有偏估计方法是近代回归分析的常用方法．本文研究了几种常用的有偏估计方法，澄清了这些方法的区别和联系．对有偏估计的一些关键点进行研究，给出了一种新的岭参数确定法和一种新的主成分概念，并讨论了这些方法的优良性．为了提高有偏估计的效率，提出了用比例因子规范模型的方法．最后，给出了说明本文方法的数值例子．     

Discrete linear bilevel programming problem

In this paper, we analyze some properties of the discrete linear bilevel program for different discretizations of the set of variables. We study the geometry of the feasible set and discuss the existence of an optimal solution. We also establish equivalences between different classes of discrete linear bilevel programs and particular linear multilevel programming problems. These equivalences are based on concave penalty functions and can be used to design penalty function methods for the solution of discrete linear bilevel programs.Support of this work has been provided by the INIC (Portugal) under Contract 89/EXA/5, by INVOTAN, FLAD, and CCLA (Portugal), and by FCAR (Québec), NSERC, and DND-ARP (Canada).     

On homogeneous and self-dual algorithms for LCP

We present some generalizations of a homogeneous and self-dual linear programming (LP) algorithm to solving the monotone linear complementarity problem (LCP). Again, while it achieves the best known interior-point iteration complexity, the algorithm does not need to use any “big-M” number, and it detects LCP infeasibility by generating a certificate. To our knowledge, this is the first interior-point and infeasible-starting algorithm for the LCP with these desired features. Research supported in part by NSF Grant DDM-9207347, the University of Iowa Oberman Fellowship and the Iowa College of Business Administration Summer Grant. Part of this work is done while the author is visiting the Delft Optimization Center at the University of Technology, Delft, Netherlands, supported by the Dutch Organization for Scientific Research (NWO).     

Solving staircase linear programs by the simplex method, 2: Pricing

This and a companion paper consider how current implementations of the simplex method may be adapted to better solve linear programs that have a staged, or ‘staircase’, structure. The preceding paper considered ‘inversion’ routines that factorize the basis and solve linear systems. The present paper examines ‘pricing’ routines that compute reduced costs for nonbasic variables and that select a variable to enter the basis at each iteration. Both papers describe extensive (although preliminary) computer experiments, and can point to some quite promising results. For pricing in particular, staircase computation strategies appear to offer modest but consistent savings; staircase selection strategies, properly chosen, may offer substantial savings in number of iterations, time per iteration, or both.     

Spontaneous magnetization of the Ising model on a layered square lattice

We have studied the Ising model on a layered square lattice with four different coupling constants and two different magnetic moments. The partition function at zero magnetic field is derived exactly. We propose a formula for the spontaneous magnetization which agrees with the exact low-temperature series expansion up to the 16th order and reduces to the exact result of Au-Yang and McCoy in a special case.