Equilibrium problems with lower and upper bounds

In this paper, we obtain some existence results of equilibrium problems with lower and upper bounds by employing a fixed-point theorem due to Ansari and Yao [1] and Ky Fan Lemma [2], respectively. Our results give answers to the open problem raised by Isac, Sehgal and Singh [3].     

Upper and lower bounds for sloshing frequencies by intermediate problems

This article presents very good, rigorous, numerical estimates for the sloshing eigenvalues of two families of two-dimensional regions. For each region the eigenvalues are proportional to the square of the frequencies of small lateral oscillations of an ideal fluid under the influence of gravity in a canal or in a horizontal cylindrical tank that has the region as cross-section. Our estimates are obtained by using conformal transformations that carry rectangles onto the regions, and then by employing intermediate problems with a generalized special choice to find upper and lower bounds to eigenvalues. The transformed problems are analogous to certain nonuniform string eigenvalue problems we estimate in a similar way.

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Résumé On calcule des bornes inférieures et supérieures aux fréquences d'oscillation latérale d'un liquide idéal sous l'influence de la gravité dans un réservoir cylindrique ou dans un canal. Les carrés des fréquences sont proportionnnels aux valeurs propres d'un opérateur aux dérivées partielles dans un ensemble de deux dimensions qui prend la forme de la section du liquide. Pour déterminer les bornes aux valeurs propres, on se sert des fonctions analytiques qui transforment les sections en rectangles ainsi que de problèmes intermédiaires choisis spécialement sous un aspect général. Les problèmes aux valeurs propres résultant des transformations sont analogues à certains problèmes de cordes vibrantes pour lesquels on peut calculer des bornes aux valeurs propres de façon semblable.

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This work was supported by the Department of the Navy, Naval Sea Command under Contract No. N 00024-81-C-5301.     

New lower bounds for bin packing problems with conflicts

The bin packing problem with conflicts (BPC) consists of minimizing the number of bins used to pack a set of items, where some items cannot be packed together in the same bin due to compatibility restrictions. The concepts of dual-feasible functions (DFF) and data-dependent dual-feasible functions (DDFF) have been used in the literature to improve the resolution of several cutting and packing problems. In this paper, we propose a general framework for deriving new DDFF as well as a new concept of generalized data-dependent dual-feasible functions (GDDFF), a conflict generalization of DDFF. The GDDFF take into account the structure of the conflict graph using the techniques of graph triangulation and tree-decomposition. Then we show how these techniques can be used in order to improve the existing lower bounds.     

Weighted quasi-equilibrium problems with lower and upper bounds

This paper is devoted to studying the solution existence of weighted quasi-equilibrium problems with lower and upper bounds by using maximal element theorems, a fixed point theorem of set-valued mappings and Fan–KKM theorem, respectively. Some new results are obtained.     

New classes of fast lower bounds for bin packing problems

The bin packing problem is one of the classical NP-hard optimization problems. In this paper, we present a simple generic approach for obtaining new fast lower bounds, based on dual feasible functions. Worst-case analysis as well as computational results show that one of our classes clearly outperforms the previous best “economical” lower bound for the bin packing problem by Martello and Toth, which can be understood as a special case. In particular, we prove an asymptotic worst-case performance of 3/4 for a bound that can be computed in linear time for items sorted by size. In addition, our approach provides a general framework for establishing new bounds. Received: August 11, 1998 / Accepted: February 1, 2001?Published online September 17, 2001     

Systems of quasi-equilibrium problems with lower and upper bounds

In this work, we consider systems of quasi-equilibrium problems with lower and upper bounds and establish the existence of their solutions by using some known maximal element theorems for a family of multivalued maps. Our problems are more general than the one posed in [G. Isac, V.M. Sehgal, S.P. Singh, An alternative version of a variational inequality, Indian J. Math. 41 (1999) 25–31]. As a particular case, we also get the answer to the problem raised in [G. Isac, V.M. Sehgal, S.P. Singh, An alternative version of a variational inequality, Indian J. Math. 41 (1999) 25–31].     

A new criterion for tractability of multivariate problems

Tractability of multivariate problems studies their complexity with respect to the number of variables, d$d$, and the accuracy of the solution ε$\epsilon$. Different types of tractability have been used, such as polynomial tractability and weak tractability and others. These tractability types, however, do not express the complexity with respect to the number of bits of accuracy.     

Upper and lower bounds for solutions of generalized two-point boundary value problems

This paper is concerned with two-point boundary value problems for systems of differential equations and integro-differential equations. If ?, ψ and Φ, Ψ are functions which satisfy certain differential (integro-differential) inequalities, then the given problem has a solutionu ^* such that ?≦u ^*≦ψ and Φ≦u ^*′≦Ψ.     

Upper and lower bounds for graph-diameter problems with application to record allocation

This paper studies three graph problems with parameters n, the number of nodes, e, the number of edges, and k, the diameter of the graph. Given any two of these three parameters, the problem is to construct a directed graph which minimizes or maximizes the third. The first problem has its origin in a recent study of record allocation in a paged computer system. It is shown how to construct graphs that are optimal for all three problems in some cases and are asymptotically optimal for other cases. The solution of the second problem answers a question raised by Berge in “The Theory of Graphs and its Application,” 1962.     

A computational analysis of lower bounds for big bucket production planning problems

In this paper, we analyze a variety of approaches to obtain lower bounds for multi-level production planning problems with big bucket capacities, i.e., problems in which multiple items compete for the same resources. We give an extensive survey of both known and new methods, and also establish relationships between some of these methods that, to our knowledge, have not been presented before. As will be highlighted, understanding the substructures of difficult problems provide crucial insights on why these problems are hard to solve, and this is addressed by a thorough analysis in the paper. We conclude with computational results on a variety of widely used test sets, and a discussion of future research.     

On lower iteration complexity bounds for the convex concave saddle point problems

Mathematical Programming - In this paper, we study the lower iteration complexity bounds for finding the saddle point of a strongly convex and strongly concave saddle point problem: $$min _xmax...     

Simple characterizations of exponential tractability for linear multivariate problems

We study linear multivariate problems defined as the approximation of compact linear multivariate operators over Hilbert spaces. We provide necessary and sufficient conditions on various notions of tractability. These conditions are mainly given in terms of sums of certain functions depending on the singular values of the multivariate problem. In particular, most of these conditions do not require the ordering of these singular values, which in many cases is difficult to achieve.     

Computing closely matching upper and lower bounds on textile nesting problems

We consider an industrial cutting problem in textile manufacturing and report on algorithms for computing cutting images and lower bounds on waste for this problem. For the upper bounds we use greedy strategies based on hodographs and global optimization based on simulated annealing. For the lower bounds we use branch-and-bound methods for computing optimal solutions of placement subproblems that determine the performance of the overall subproblem. The upper bounds are computed in less than an hour on a common-day workstation and are competitive in quality with results obtained by human nesters. The lower bounds take a few hours to compute and are within 0.4% of the upper bound for certain types of clothing (e.g., for pants).     

The error bounds and tractability of quasi-Monte Carlo algorithms in infinite dimension

Dimensionally unbounded problems are frequently encountered in practice, such as in simulations of stochastic processes, in particle and light transport problems and in the problems of mathematical finance. This paper considers quasi-Monte Carlo integration algorithms for weighted classes of functions of infinitely many variables, in which the dependence of functions on successive variables is increasingly limited. The dependence is modeled by a sequence of weights. The integrands belong to rather general reproducing kernel Hilbert spaces that can be decomposed as the direct sum of a series of their subspaces, each subspace containing functions of only a finite number of variables. The theory of reproducing kernels is used to derive a quadrature error bound, which is the product of two terms: the generalized discrepancy and the generalized variation.

Tractability means that the minimal number of function evaluations needed to reduce the initial integration error by a factor is bounded by for some exponent and some positive constant . The -exponent of tractability is defined as the smallest power of in these bounds. It is shown by using Monte Carlo quadrature that the -exponent is no greater than 2 for these weighted classes of integrands. Under a somewhat stronger assumption on the weights and for a popular choice of the reproducing kernel it is shown constructively using the Halton sequence that the -exponent of tractability is 1, which implies that infinite dimensional integration is no harder than one-dimensional integration.

     

Upper and lower bounds for eigenvalues of torsion and bending problems by finite difference methods

Zusammenfassung In diesem Artikel werden mit Hilfe von Differenzenanalogien mathematisch strenge obere und untere Schranken für die Torsions- und Biegungsfrequenzen freier Balken hergeleitet. Zur Lösung des Differenzenproblems werden sodann Sturm-Folgen Algorithmen vorgeschlagen. Tabellen numerischer Ergebnisse für verschiedene Beispiele mit stückweise konstanter Masse und Biegesteifigkeit zeigen eine Überlegenheit gegenüber Ergebnissen, die früher mit Hilfe von Raleigh-Ritzschen- und Vergleichsoperator-Verfahren erhalten worden waren.

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This work supported by the Department of the Navy, Bureau of Naval Weapons, under Contract NOw-62-0604-c.