A o(n logn) algorithm for LP knapsacks with GUB constraints

A specialization of the dual simplex method is developed for solving the linear programming (LP) knapsack problem subject to generalized upper bound (GUB) constraints. The LP/GUB knapsack problem is of interest both for solving more general LP problems by the dual simplex method, and for applying surrogate constraint strategies to the solution of 0–1 Multiple Choice integer programming problems. We provide computational bounds for our method of o(n logn), wheren is the total number of problem variables. These bounds reduce the previous best estimate of the order of complexity of the LP/GUB knapsack problem (due to Witzgall) and provide connections to computational bounds for the ordinary knapsack problem.We further provide a variant of our method which has only slightly inferior worst case bounds, yet which is ideally suited to solving integer multiple choice problems due to its ability to post-optimize while retaining variables otherwise weeded out by convex dominance criteria.     

On the Bartels—Golub decomposition for linear programming bases

Many implementations of the simplex method require the row of the inverse of the basis matrixB corresponding to the pivot row at each iteration for updating either a pricing vector or the nonbasic reduced costs. In this note we show that if the Bartels—Golub algorithm [1, 2] or one of its variants is used to update theLU factorization ofB, then less computing is needed if one works with the factors of the updatedB than with those ofB. These results are discussed as they apply to the column selection algorithms recently proposed by Goldfarb and Reid [4, 5] and Harris [6].This research was supported in part by the National Science Foundation under Grant GJ 36472.     

The simplex method and unrestricted variables

Suppose that the simplex method is applied to a linear programming problem havingm equality constraints andr unrestricted variables. We give a method of performing the steps of the simplex method which reduces the arithmetic operation count byrm at each iteration. This savings in operations is achieved, since the method does not update the rows of the basis inverse associated with the unrestricted variables. Similar computational savings are achieved when the method is applied to the updating of anLU-factorization of the basis matrix.This research was supported by the Natural Sciences and Engineering Research Council of Canada under Grant No. A8189 and by a postgraduate fellowship.     

Alternative dual frames for digital-to-analog conversion in sigma–delta quantization

We design alternative dual frames for linearly reconstructing signals from sigma–delta (ΣΔ) quantized finite frame coefficients. In the setting of sampling expansions for bandlimited functions, it is known that a stable rth order sigma–delta quantizer produces approximations where the approximation error is at most of order 1 / λ ^r , and λ > 1 is the oversampling ratio. We show that the counterpart of this result is not true for several families of redundant finite frames for \mathbbR^d\mathbb{R}^d when the canonical dual frame is used in linear reconstruction. As a remedy, we construct alternative dual frame sequences which enable an rth order sigma–delta quantizer to achieve approximation error of order 1/N ^r for certain sequences of frames where N is the frame size. We also present several numerical examples regarding the constructions.     

Implicit representation of generalized variable upper bounds in linear programming

In certain linear programs, especially those derived from integer programs, large numbers of constraints may have very simple form. Examples are:x _ij 1 (simple upper bounds [SUB]), _i x _ij = 1 (generalized upper bounds [GUB]) andx _ij y _i (variable upper bounds [VUB]). A class of constraints called generalized VUB [GVUB] is introduced which includes GUB and VUB as special cases. Also introduced is a method for representing GVUB constraints implicitly within the mechanics of the simplex method.Research supported in part by the Mobil Oil Corporation.     

关于图值随机元序列的强极限定理

本文研究在局部连通图中取值的随机元的r-阶均值集、r-阶广义样本均值集的基本性质及其极限性质.利用关于随机元的分离引理以及随机游动的常返性,得到了关于图值随机元序列广义强大数定理.推广了已有的结果.     

The asymptotic number of integer stochastic matrices

Let Hⁿ_r be the number of n × n matrices, with nonnegative integer elements, all of whose row and column sums are equal to some prescribed integer r. Similarly, let Aⁿ_r be the number of n × n (0.1) matrices with common row and column sum r. An asymptotic formula for Hⁿ_r is stated and proved, the method of proof being essentially elementary. A simple modification of the proof yields an analogous asymptotic formula for Aⁿ_r. The latter agrees with a result of O'Neil, obtained by a completely different method.     

Stable modification of explicitLU factors for simplex updates

This paper presents a method for modifying anLU factorization of a matrixA when a Simplex update is performed onA. At each stage of the algorithm, two options are allowed, and in one of these a row permutation ofL is made so that theLU structure is always maintained. The expanding file of operators used in other methods is not required. The result, of an error analysis is presented in which the error is bounded by the growth in the partial sums ofLU. Various pivotal strategies which minimize a growth bound are discussed and an example is given in which the worst case for growth occurs repeatedly. Numerical results from experiments on ill-conditioned problems show no tendency for error growth.     

Limited Memory Space Dilation and Reduction Algorithms

In this paper, we present variants of Shor and Zhurbenko's r-algorithm, motivated by the memoryless and limited memory updates for differentiable quasi-Newton methods. This well known r-algorithm, which employs a space dilation strategy in the direction of the difference between two successive subgradients, is recognized as being one of the most effective procedures for solving nondifferentiable optimization problems. However, the method needs to store the space dilation matrix and update it at every iteration, resulting in a substantial computational burden for large-sized problems. To circumvent this difficulty, we first propose a memoryless update scheme, which under a suitable choice of parameters, yields a direction of motion that turns out to be a convex combination of two successive anti-subgradients. Moreover, in the space transformation sense, the new update scheme can be viewed as a combination of space dilation and reduction operations. We prove convergence of this new method, and demonstrate how it can be used in conjunction with a variable target value method that allows a practical, convergent implementation of the method. We also examine a memoryless variant that uses a fixed dilation parameter instead of varying degrees of dilation and/or reduction as in the former algorithm, as well as another variant that examines a two-step limited memory update. These variants are tested along with Shor's r-algorithm and also a modified version of a related algorithm due to Polyak that employs a projection onto a pair of Kelley's cutting planes. We use a set of standard test problems from the literature as well as randomly generated dual transportation and assignment problems in our computational experiments. The results exhibit that the proposed space dilation and reduction method and the modification of Polyak's method are competitive, and offer a substantial advantage over the r-algorithm and over the other tested limited memory variants with respect to accuracy as well as effort.     

A decomposition algorithm for linear relaxation of the weightedr-covering problem

In this paper the linear relaxation of the weightedr-covering problem (r-LCP) is considered. The dual problem (c-LMP) is the linear relaxation of the well-knownc-matching problem and hence can be solved in polynomial time. However, we describe a simple, but nonpolynomial algorithm in which ther-LCP is decomposed into a sequence of 1-LCP’s and its optimal solution is obtained by adding the optimal solutions of these 1-LCP’s. An 1-LCP can be solved in polynomial time by solving its dual as a max-flow problem on a bipartite graph. An accelerated algorithm based on this decomposition scheme to solve ar-LCP is also developed and its average case behaviour is studied.     

Sobolev derivatives and Itô decomposition formula for Gaussian processes using properties of their RKHSs

Malliavin calculus has been extensively developed for abstract Wiener spaces. It is of interest to develop the basic concepts of an infinite-dimensional calculus for an arbitrary Gaussian processX=(X_t), wheret T (T being a multiparameter set or, more generally, a complete separable metric space), bringing into evidence the properties of the covariance kernel (or, equivalently, the reproducing kernel Hilbert space) ofX. In this paper a definition of thekth Sobolev derivative is given and thekth chaos expansion of a functional is shown to be thekth-order divergence operator. An extension of Itô's decomposition formula is derived.     

On extremal problems of graphs and generalized graphs

Anr-graph is a graph whose basic elements are its vertices and r-tuples. It is proved that to everyl andr there is anε(l, r) so that forn>n ₀ everyr-graph ofn vertices andn ^{r−ε(l, r)} r-tuples containsr. l verticesx ^(j), 1≦j≦r, 1≦i≦l, so that all ther-tuples occur in ther-graph.     

Restart procedures for the conjugate gradient method

A compact and flexible updating procedure using matrix augmentation is developed. It is shown that the representation of the updated inverse does not grow monotonically in size, and may actually decrease during certain simplex iterations. Angular structures, such as GUB, are handled naturally within the partitioning framework, and require no modifications of the simplex method.     

An exact formula for the move-to-front rule for self-organizing lists

A collection of items (e.g., books), each with an associated weight (or popularity), is arranged in a row. At each unit of time an item is removed with probability proportional to its weight and replaced at the left end of the row. Thismove-to-front rule gives a Markov chain on permutations often known as theTsetlin library. We derive an exact and tractable formula for the probability of any permutation after any number of moves. From the formula we read off previously studied quantities of interest associated with the chain, such as the stationary distribution and eigenvalues. Measuring discrepancy from stationarity by separation, we use the formula to find the initial arrangement giving the slowest convergence to stationarity. The time to stationarity in this case is a convolution of geometric random variables which we analyze for three natural choices of weights. We also assess the time required for an important functional, namely, expected search cost, to approach its stationary value.     

The most-obtuse-angle row pivot rule for achieving dual feasibility: A computational study

We recently proposed several new pivot rules for achieving dual feasibility in linear programming, which are distinct from existing ones: the objective function value will no longer change necessarily monotonically in their solution process. In this paper, we report our further computational testing with one of them, the most-obtuse-angle rule. A two-phase dual simplex algorithm, in which the rule is used as a row selection rule for Phase-1, has been implemented in FORTRAN 77 modules, and was tested on a set of standard linear programming problems from NETLIB. We show that, if full pricing is applied, our code unambiguously will outperform MINOS 5.3, one of the best implementations of the simplex algorithm at present.     

Efficient dual simplex algorithms for the assignment problem

Efficient algorithms based upon Balinski's signature method are described for solving then × n assignment problem. These algorithms are special variants of the dual simplex method and are shown to have computational bounds of O(n ³). Variants for solving sparse assignment problems withm arcs that require O(m) space and O(mn + n ² logn) time in the worst case are also presented.This research was supported in part by the National Science Foundation under Grant No. MCS-8006064 and by the Army Research Office under Contracts No. DAAG 29-82-K0163 and DAAG 29-83-K0106     

Basis properties and algorithmic specializations for gub transformed networks

Applying standard transformations of generalized upper bounding (GUB) theory to a pure or generalized network basis is shown to yield a reduced working basis that is itself a basis for a reduced network. As a result, the working basis can be represented via specialized data structures for networks. The resultant GUB based specializations to the network simplex algorithm are described.     

Three geometric inequalities for a simplex

In this paper, we obtain three new geometric inequalities for ann-dimensional simplex in then-dimensional Euclidean spaceE ⁿ . As special cases we find two known inequalities from L. Fejes Tóth and M. S. Klamkin, respectively.     

Error estimates for spline interpolants on equidistant grids

Summary For oddm, the error of them-th-degree spline interpolant of power growth on an equidistant grid is estimated. The method is based on a decomposition formula for the spline function, which locally can be represented as an interpolation polynomial of degreem which is corrected by an (m+1)-st.-order difference term.Dedicated to Prof. Dr. Karl Zeller on the occasion of his 60th birthday     

线性规划的对偶基线算法

In this paper,we studied the dual form of the basic line algorthm for linear programs.It can be easily implemented in tableau that similar to the primal/dual simplex method.Different from primal simplex method or dual simplex method,the dual basic line algorithm can keep primal feasibility and dual feasibility at the same time in a tableau,which makes it more efficient than the former ones.Principles and convergence of dual basic line algorthm were discussed.Some examplex and computational experience were given to illustrate the efficiency of our method.