Worst-case analysis of a dynamic channel assignment strategy

We consider the problem of channel assignment in cellular networks with arbitrary reuse distance. We show upper and lower bounds for the competitive ratio of a previously proposed and widely studied version of dynamic channel assignment, which we refer to as the greedy algorithm. We study two versions of this algorithm: one that performs reassignment of channels, and one that never reassigns channels to calls. For reuse distance 2, we show tight bounds on the competitive ratio of both versions of the algorithm. For reuse distance 3, we show non-trivial lower bounds for both versions of the algorithm.     

Equality conditions for lower bounds on the smallest singular value of a bidiagonal matrix

Several lower bounds have been proposed for the smallest singular value of a square matrix, such as Johnson’s bound, Brauer-type bound, Li’s bound and Ostrowski-type bound. In this paper, we focus on a bidiagonal matrix and investigate the equality conditions for these bounds. We show that the former three bounds give strict lower bounds if all the bidiagonal elements are non-zero. For the Ostrowski-type bound, we present an easily verifiable necessary and sufficient condition for the equality to hold.     

A Comparison of homogenization, Hashin-Shtrikman bounds and the Halpin-Tsai Equations

In this paper we study a unidirectional and elastic fiber composite. We use the homogenization method to obtain numerical results of the plane strain bulk modulus and the transverse shear modulus. The results are compared with the Hashin-Shtrikman bounds and are found to be close to the lower bounds in both cases. This indicates that the lower bounds might be used as a first approximation of the plane strain bulk modulus and the transverse shear modulus. We also point out the connection with the Hashin-Shtrikman bounds and the Halpin-Tsai equations. Optimal bounds on the fitting parameters in the Halpin-Tsai equations have been formulated.     

Lower Bounds for the Rate of Convergence of Dynamic Reconstruction Algorithms for Distributed-Parameter Systems

We obtain lower bounds for the rate of convergence of reconstruction algorithms for distributed-parameter systems of parabolic type. In the case of a pointwise constraint on control for known reconstruction algorithms, we establish a lower bound on the rate of convergence, which shows that, given certain conditions, for each solution of the system one can choose such a collection of measurements so that the reconstruction error will not be less than a certain value. In the case of unbounded controls, we obtain lower bounds for a possible reconstruction error for each trajectory as well as for a given set of trajectories. For a system of special form, we construct an algorithm for which we obtain upper and lower bounds for accuracy having identical order for a specific choice of matching of the parameters.     

Tail bounds for the joint distribution of the surplus prior to and at ruin

For the classical risk model with Poisson arrivals, we study the (bivariate) tail of the joint distribution of the surplus prior to and at ruin. We obtain some exact expressions and new bounds for this tail, and we suggest three numerical methods that may yield upper and lower bounds for it. As a by-product of the analysis, we obtain new upper and lower bounds for the probability and severity of ruin. Many of the bounds in the present paper improve and generalise corresponding bounds that have appeared earlier. For the numerical bounds, their performance is also compared against bounds available in the literature.     

Heat kernels of non-symmetric jump processes with exponentially decaying jumping kernel

In this paper we study the transition densities for a large class of non-symmetric Markov processes whose jumping kernels decay exponentially or subexponentially. We obtain their upper bounds which also decay at the same rate as their jumping kernels. When the lower bounds of jumping kernels satisfy the weak upper scaling condition at zero, we also establish lower bounds for the transition densities, which are sharp.     

Reducing model risk via positive and negative dependence assumptions

We give analytical bounds on the Value-at-Risk and on convex risk measures for a portfolio of random variables with fixed marginal distributions under an additional positive dependence structure. We show that assuming positive dependence information in our model leads to reduced dependence uncertainty spreads compared to the case where only marginals information is known. In more detail, we show that in our model the assumption of a positive dependence structure improves the best-possible lower estimate of a risk measure, while leaving unchanged its worst-possible upper risk bounds. In a similar way, we derive for convex risk measures that the assumption of a negative dependence structure leads to improved upper bounds for the risk while it does not help to increase the lower risk bounds in an essential way. As a result we find that additional assumptions on the dependence structure may result in essentially improved risk bounds.     

Ranking lower bounds for the bin-packing problem

In this paper, we evaluate different known lower bounds for the bin-packing problem including linear programming relaxation (LP), Lagrangean relaxation (LR), Lagrangean decomposition (LD) and the Martello–Toth (MT) [Martello, S., Toth, P., Knapsack Problems: Algorithms and Computer Implementations, Wiley, New York, 1990] lower bounds. We give conditions under which Lagrangean bounds are superior to the LP bound, show that Lagrangean decomposition (LD) yields the same bound as Lagrangean relaxation (LR) and prove that the MT lower bound is a Lagrangean bound evaluated at a finite set of Lagrange multipliers; hence, it is no better than the LR and LD lower bounds.     

The effect of transformations on the approximation of univariate (convex) functions with applications to Pareto curves

In the literature, methods for the construction of piecewise linear upper and lower bounds for the approximation of univariate convex functions have been proposed. We study the effect of the use of transformations on the approximation of univariate (convex) functions. In this paper, we show that these transformations can be used to construct upper and lower bounds for nonconvex functions. Moreover, we show that by using such transformations of the input variable or the output variable, we obtain tighter upper and lower bounds for the approximation of convex functions than without these approximations. We show that these transformations can be applied to the approximation of a (convex) Pareto curve that is associated with a (convex) bi-objective optimization problem.     

Fractal methods and the problem of estimating scaling exponents: A new approach based on upper and lower linear bounds

The characterization of irregular objects with fractal methods often leads to the estimation of the slope of a function which is plotted versus a scale parameter. The slope is usually obtained with a linear regression. The problem is that the fit is usually not acceptable from the statistical standpoint. We propose a new approach in which we use two straight lines to bound the data from above and from below. We call these lines the upper and lower linear bounds. We propose to define these bounds as the solution of an optimization problem. We discuss the solution of this problem and we give an algorithm to obtain its solution. We use the difference between the upper and lower linear bounds to define a measure of the degree of linearity in the scaling range. We illustrate our method by analyzing the fluctuations of the variogram in a microresistivity well log from an oil reservoir in the North Sea.     

Improved lower bounds for the early/tardy scheduling problem with no idle time

In this paper, we consider the single machine earliness/tardiness scheduling problem with no idle time. Two of the lower bounds previously developed for this problem are based on Lagrangean relaxation and the multiplier adjustment method, and require an initial sequence. We investigate the sensitivity of the lower bounds to the initial sequence, and experiment with different dispatch rules and some dominance conditions. The computational results show that it is possible to obtain improved lower bounds by using a better initial sequence. The lower bounds are also incorporated in a branch-and-bound algorithm, and the computational tests show that one of the new lower bounds has the best performance for larger instances.     

On the complexity of parabolic initial-value problems with variable drift

We study the intrinsic difficulty of solving linear parabolic initial-value problems numerically at a single point. We present a worst-case analysis for deterministic as well as for randomized (or Monte Carlo) algorithms, assuming that the drift coefficients and the potential vary in given function spaces. We use fundamental solutions (parametrix method) for equations with unbounded coefficients to relate the initial-value problem to multivariate integration and weighted approximation problems. Hereby we derive lower and upper bounds for the minimal errors. The upper bounds are achieved by algorithms that use Smolyak formulas and, in the randomized case, variance reduction. We apply our general results to equations with coefficients from Hölder classes, and here, in many cases, the upper and lower bounds almost coincide and our algorithms are almost optimal.     

Convex relaxations of chance constrained optimization problems

In this paper we develop convex relaxations of chance constrained optimization problems in order to obtain lower bounds on the optimal value. Unlike existing statistical lower bounding techniques, our approach is designed to provide deterministic lower bounds. We show that a version of the proposed scheme leads to a tractable convex relaxation when the chance constraint function is affine with respect to the underlying random vector and the random vector has independent components. We also propose an iterative improvement scheme for refining the bounds.     

Maximum norms of random sums and transient pattern formation

Many interesting and complicated patterns in the applied sciences are formed through transient pattern formation processes. In this paper we concentrate on the phenomenon of spinodal decomposition in metal alloys as described by the Cahn-Hilliard equation. This model depends on a small parameter, and one is generally interested in establishing sharp lower bounds on the amplitudes of the patterns as the parameter approaches zero. Recent results on spinodal decomposition have produced such lower bounds. Unfortunately, for higher-dimensional base domains these bounds are orders of magnitude smaller than what one would expect from simulations and experiments. The bounds exhibit a dependence on the dimension of the domain, which from a theoretical point of view seemed unavoidable, but which could not be observed in practice.

In this paper we resolve this apparent paradox. By employing probabilistic methods, we can improve the lower bounds for certain domains and remove the dimension dependence. We thereby obtain optimal results which close the gap between analytical methods and numerical observations, and provide more insight into the nature of the decomposition process. We also indicate how our results can be adapted to other situations.

     

Bounding Contingent Claim Prices via Hedging Strategy with Coherent Risk Measures

We generalize the notion of arbitrage based on the coherent risk measure, and investigate a mathematical optimization approach for tightening the lower and upper bounds of the price of contingent claims in incomplete markets. Due to the dual representation of coherent risk measures, the lower and upper bounds of price are located by solving a pair of semi-infinite linear optimization problems, which further reduce to linear optimization when conditional value-at-risk (CVaR) is used as risk measure. We also show that the hedging portfolio problem is viewed as a robust optimization problem. Tuning the parameter of the risk measure, we demonstrate by numerical examples that the two bounds approach to each other and converge to a price that is fair in the sense that seller and buyer face the same amount of risk.     

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).     

Some upper and lower bounds on PSD-rank

Positive semidefinite rank (PSD-rank) is a relatively new complexity measure on matrices, with applications to combinatorial optimization and communication complexity. We first study several basic properties of PSD-rank, and then develop new techniques for showing lower bounds on the PSD-rank. All of these bounds are based on viewing a positive semidefinite factorization of a matrix M as a quantum communication protocol. These lower bounds depend on the entries of the matrix and not only on its support (the zero/nonzero pattern), overcoming a limitation of some previous techniques. We compare these new lower bounds with known bounds, and give examples where the new ones are better. As an application we determine the PSD-rank of (approximations of) some common matrices.     

Upper and Lower Bounds for the Lengths of Steiner Trees in 3-Space

We present a method of determining upper and lower bounds for the length of a Steiner minimal tree in 3-space whose topology is a given full Steiner topology, or a degenerate form of that full Steiner topology. The bounds are tight, in the sense that they are exactly satisfied for some configurations. This represents the first nontrivial lower bound to appear in the literature. The bounds are developed by first studying properties of Simpson lines in both two and three dimensional space, and then introducing a class of easily constructed trees, called midpoint trees, which provide the upper and lower bounds. These bounds can be constructed in quadratic time. Finally, we discuss strategies for improving the lower bound.Supported by a grant from the Australia Research Council.     

Comparison of formulations and a heuristic for packing Steiner trees in a graph

In this paper, we consider the problem of packing Steiner trees in a graph. This problem arises during the global routing phase of circuit layout design. We consider various integer programming formulations and rank them according to lower bounds they provide as LP-relaxations. We discuss a solution procedure to obtain both lower and upper bounds using one of the LP-relaxations. Computational results to test the effectiveness of our procedures are provided.