Probabilistic properties of the dual structure of the multidimensional knapsack problem and fast statistically efficient algorithms

Properties of several dual characteristics of the multidimensional knapsack problem (such as the probability of existence of-optimal and optimal-feasible Lagrange function generalized saddle points, magnitude of relative duality gap, etc.) are investigated for different probabilistic models. Sufficient conditions of good asymptotic behavior of the dual characteristics are given. A fast statistically efficient approximate algorithm with linear running time complexity for problems with random coefficients is presented.This paper was written when the author was affiliated with Chelyabinsk State Technical University and the Moscow Physical and Technical Institute, Russia.     

Analysis of a class of fractional programming problems

We propose a solution strategy for fractional programming problems of the form max _xx g(x)/ (u(x)), where the function satisfies certain convexity conditions. It is shown that subject to these conditions optimal solutions to this problem can be obtained from the solution of the problem max _xx g(x) + u(x), where is an exogenous parameter. The proposed strategy combines fractional programming andc-programming techniques. A maximal mean-standard deviation ratio problem is solved to illustrate the strategy in action.     

Semiparametric Sufficient Efficiency Conditions and Duality Models for Multiobjective Fractional Subset Programming Problems with Generalized (Ф, ρ, θ)-Convex Functions

In this paper, we present a multitude of global semiparametric sufficient efficiency and duality results under generalized (, , )-convexity assumptions for a multiobjective fractional subset programming problem.     

An interpolation method for the solution of identification problems for a multidimensional functional system described by a Urysohn operator

An algorithm is constructed for identifying deterministic dynamical systems described using a Urysohn operator, i. e., the investigation is conducted under the hypothesis that the investigated object has a partially known structure, a grey box, and the identification problem consists in approximating the Urysohn nonlinearity. The initial information for solving the identification problem is the aggregate of the input effects on a physical object, the input, and also its reaction to these effects, the output. The accuracy of the algorithm is discussed.Translated from Vychislitel'naya i prikladnaya matematika, No. 70, pp. 31–36, 1990.     

Identification of the Optimal Active Set in a Noninterior Continuation Method for LCP

This paper concerns about the possibility of identifying the active set in a noninterior continuation method for solving the standard linear complementarity problem based on the algorithm and theory presented by Burke and Xu (J. Optim. Theory Appl. 112 (2002) 53). It is shown that under the assumptions of P-matrix and nondegeneracy, the algorithm requires at most Olog(₀₀/)) iterations to find the optimal active set, where ₀ is the width of the neighborhood which depends on the initial point, ₀> 0 is the initial smoothing parameter, is a positive number which depends on the problem and the initial point, and is a small positive number which depends only on the problem.     

Factorizations of the Complete Graph into C 5-Factors and 1-Factors

In this paper, we show that K_10n can be factored into C₅-factors and 1-factors for all non-negative integers and satisfying 2+=10n–1.Research partially supported by an NSF-AWM Mentoring Travel Grant     

Quadratically constrained minimum cross-entropy analysis

Quadratically constrained minimum cross-entropy problem has recently been studied by Zhang and Brockett through an elaborately constructed dual. In this paper, we take a geometric programming approach to analyze this problem. Unlike Zhang and Brockett, we separate the probability constraint from general quadratic constraints and use two simple geometric inequalities to derive its dual problem. Furthermore, by using the dual perturbation method, we directly prove the strong duality theorem and derive a dual-to-primal conversion formula. As a by-product, the perturbation proof gives us insights to develop a computation procedure that avoids dual non-differentiability and allows us to use a general purpose optimizer to find an-optimal solution for the quadratically constrained minimum cross-entropy analysis.     

Global Optimization Algorithm for the Nonlinear Sum of Ratios Problem

This article presents a branch-and-bound algorithm for globally solving the nonlinear sum of ratios problem (P). The algorithm economizes the required computations by conducting the branch-and-bound search in ^p, rather than in ⁿ, where p is the number of ratios in the objective function of problem (P) and n is the number of decision variables in problem (P). To implement the algorithm, the main computations involve solving a sequence of convex programming problems for which standard algorithms are available.     

(LF) spaces and distributions on compact groups and spectral synthesis onR/2πZ

LetE R/2Z be closed with Lebesgue measure 0. We imbed the pseudo-measuresTsupported byE into a space of distributions on a specific compact connected group. The reason for this approach is to make use of the more tractable differentiable absolutely convergent Fourier series for the general problem to determine whenT is a measure. The specific results are outlined in the Introduction. Applications of the techniques presented here are used to obtain new criteria that a Helson set be a set of spectral synthesis in the author's forthcoming work (viz., A Support Preserving Hahn-Banach Property and the Helson-S Set Problem and The Helson-S Set Problem and Discontinuous Homomorphisms on Metric Algebras).     

The martin boundary for harmonic functions on groups of automorphisms of a homogeneous tree

Consider a closed subgroup of the automorphism group of a homogeneous treeT, and assume that acts transitively on the vertex set. Suppose that is a probability measure on which has continuous density with respect to Haar measure and whose support is compact open and generates as a closed semigroup. It is shown that the Martin boundary of with respect to the random walk with law coincides with the space of ends ofT. This extends known results for free groups and applies, for example, to the affine group over a non archimedean local field.     

Convolution semigroups and central limit theorem associated with a dual convolution structure

We consider hypergroups associated with Jacobi functions ⁽⁾ (x), (–1/2). We prove the existence of a dual convolution structure on [0,+[i(]0,s ₀]{{) =++1,s ₀=min(,–+1). Next we establish a Lévy-Khintchine type formula which permits to characterize the semigroup and the infinitely divisible probabilities associated with this dual convolution, finally we prove a central limit theorem.     

On-line k-Truck Problem and Its Competitive Algorithms

In this paper, based on the Position Maintaining Strategy (PMS for short), on-line scheduling of k-truck problem, which is a generalization of the famous k-server problem, is originally presented by our team. We proposed several competitive algorithms applicable under different conditions for solving the on-line k-truck problem. First, a competitive algorithm with competitive ratio 2k+1/ is given for any 1. Following that, if (c+1)/(c-1) holds, then there must exist a (2k-1)-competitive algorithm for k-truck problem, where c is the competitive ratio of the on-line algorithm about the relevant k-server problem. And then a greedy algorithm with competitive ratio 1+/, where lambda is a parameter related to the structure property of a given graph, is given. Finally, competitive algorithms with ratios 1+1/ are given for two special families of graphs.     

On generalized gamma functions related to the Laurent coefficients of the Riemann zeta function

Summary We study a class of generalized gamma functions _k (z) which relate to the generalized Euler constants _k (basically the Laurent coefficients of(s)) as (z) does to the Euler constant. A new series expansion for _k is derived, and the constant term in the asymptotic expansion for log _k (z) is studied in detail. These and related constants are numerically computed for 1 k 15.     

Stellare Abänderungen und Schälbarkeit von Komplexen und Polytopen

Brugesser and Mani proved that the boundary-complex of a convex polytope can be shelled. This result lead to McMullen's proof of the Upper-bound-conjecture. We show that the shellability of complexes has a close connection to the theory of stellar operations. Several results on special shelling procedures and on non-shellable complexes are obtained.     

A topological characterization for closed sets under polar duality in ℚ n

A topological characterization is given for closed sets in ⁿ under the restriction of (cone) polar duality to ⁿ .     

A Long-Step, Cutting Plane Algorithm for Linear and Convex Programming

A cutting plane method for linear programming is described. This method is an extension of Atkinson and Vaidya's algorithm, and uses the central trajectory. The logarithmic barrier function is used explicitly, motivated partly by the successful implementation of such algorithms. This makes it possible to maintain primal and dual iterates, thus allowing termination at will, instead of having to solve to completion. This algorithm has the same complexity (O(nL ²) iterations) as Atkinson and Vaidya's algorithm, but improves upon it in that it is a long-step version, while theirs is a short-step one in some sense. For this reason, this algorithm is computationally much more promising as well. This algorithm can be of use in solving combinatorial optimization problems with large numbers of constraints, such as the Traveling Salesman Problem.     

Existence for the Cahn-Hilliard phase separation model with a nondifferentiable energy

Summary The Cahn-Hilliard model for phase separation in a binary alloy leads to the equations (I) u_t=w, (II) w= (u)– u with an associated energy functional F(u)=f [(u)+ +¦u¦²/2] dx. In this paper we discuss the existence theory for initial bounday value problems arising from modifications to the Cahn-Hilliard model due to the addition of the non-differentiable term ¦u¦dx to the energy F(u).     

An efficient heuristic algorithm for minimum matching

We give a new heuristic algorithm for minimum matching problems and apply it to the Euclidean problem with random vertices in 2 dimensions. The algorithm is based on simulated annealing and performs in practice faster than previous heuristic algorithms yielding suboptimal solutions of the same good quality. From configurations with up toN=20.000 vertices in the unit square we estimate that the length of a minimum matching scales asymptotically asLN with (=0.3123±0.0016.

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Zusammenfassung Wir stellen einen neuen heuristischen Algorithmus für minimale Matching-Probleme vor und wenden diesen auf das euklidische Problem mit zufÄlliger Punkteverteilung in 2 Dimensionen an. Auf Simulated Annealing basierend lÄuft der Algorithmus schneller als frühere heuristische Algorithmen und erreicht dabei suboptimale Lösungen gleich guter QualitÄt. Aus Konfigurationen mit bis zuN=20.000 Punkten im Einheitsquadrat schÄtzen wir, da\ für die LÄnge des minimalen Matchings asymptotischLN mit=0.3123±0.0016 gilt.

     

Large Deviations of a Storage Process with Fractional Brownian Motion as Input

We study probabilities of large extremes of the storage process Y(t) = sup _t (X() - X(t) - c( - t)), where X(t) is the fractional Brownian motion. We derive asymptotic behavior of the maximum tail distribution for the process on fixed or slowly increased intervals by a reduction the problem to a large extremes problem for a Gaussian field.     

A branch-bound algorithm for a wiring problem

A simple computational method for solving a specific wiring problem related to the construction of the RC 4000 computer is described. The intimate relationship between the wiring problem and the traveling salesman problem is established, and the algorithm is based upon the branch and bound technique as employed by J.D.C. Little et al. [1] for solving the latter problem.Adapted from the thesis Fixed-cost and Other Network Flow Problems presented to The Institute of Mathematical Statistics and Operations Research, The Technical University of Denmark in partial fulfilment of the requirements for the licentiate degree, April 1967.