The Tolerance Approach to Sensitivity Analysis of Matrix Coefficients in Linear Programming: General Perturbations

In the tolerance approach to sensitivity analysis of matrix coefficients in a linear programme, the maximum tolerance percentage characterizes the largest percentage within which selected matrix coefficients may vary simultaneously and independently while retaining the same set of optimal basic variables. While it is difficult to calculate exactly the maximum tolerance percentage under perturbations in multiple rows or columns, we show herein how it can be approximated.     

Tolerance approach to sensitivity analysis in linear complementarity problems

In this paper, we apply the tolerance approach proposed by Wendell for sensitivity analysis in linear programs to study sensitivity analysis in linear complementarity problems. In the tolerance approach, we find the range or the maximum tolerance within which the coefficients of the right-hand side of the problem can vary simultaneously and independently such that the solution of the original and the perturbed problems have the same index set of nonzero elements.The work of the first author was completed while he was at Virginia Commonwealth University, Richmond, Virginia.     

An adaptive boundary collocation method for linear PDEs

The article proposes an adaptive algorithm based on a boundary collocation method for linear PDEs satisfying the maximal principle with possibly nonlinear boundary conditions. Given the error tolerance and an initial number of terms in the solution expansion, the algorithm computes expansion coefficients by collocation of boundary conditions and evaluates the maximum absolute error on the boundary. If error exceeds the error tolerance, additional expansion terms and boundary collocation points are added and the process repeated until the tolerance is satisfied. The performance of the algorithm is illustrated by an example of the potential flow past a cylinder placed between parallel walls. © 1995 John Wiley & Sons, Inc.     

Multiparametric sensitivity analysis in programming problem with linear-plus-linear fractional objective function

In this paper, we study multiparametric sensitivity analysis for programming problems with linear-plus-linear fractional objective function using the concept of maximum volume in the tolerance region. We construct critical regions for simultaneous and independent perturbations in the objective function coefficients and in the right-hand-side vector of the given problem. Necessary and sufficient conditions are derived to classify perturbation parameters as `focal' and `non-focal'. Non-focal parameters can have unlimited variations, because of their low sensitivity in practice, these parameters can be deleted from the analysis. For focal parameters, a maximum volume tolerance region is characterized. Theoretical results are illustrated with the help of a numerical example.     

Global optimization of fractional programs

Dinkelbach's global optimization approach for finding the global maximum of the fractional programming problem is discussed. Based on this idea, a modified algorithm is presented which provides both upper and lower bounds at each iteration. The convergence of the lower and upper bounds to the global maximum function value is shown to be superlinear. In addition, the special case of fractional programming when the ratio involves only linear or quadratic terms is considered. In this case, the algorithm is guaranteed to find the global maximum to within any specified tolerance, regardless of the definiteness of the quadratic form.     

QRS complex detection based on multi wavelet packet decomposition

We present in this paper a wavelet packet based QRS complex detection algorithm. Our proposed algorithm consists of a particular combination of two vectors obtained by applying a designed routine of QRS detection process using ‘haar’ and ‘db10’ wavelet functions respectively. The QRS complex detection routine is based on the histogram approach where our key idea was to search for the node with highest number of histogram coefficients, at center, which we assume that they are related to the iso-electric baseline whereas the remaining least number coefficients reflect the R waves peaks. Following a classical approach based of a calculated fixed threshold, the possible QRS complexes will be determined. The QRS detection complex algorithm has been applied to the whole MIT-BIH arrhythmia Database to assess its robustness. The algorithm reported a global sensitivity of 98.68%, positive predictive value of 97.24% and a percentage error of 04.12%. Eventhough, the obtained global results are not as excellent as expected, we have demonstrate that our designed QRS detection algorithm performs good on a partial selected high percentage of the whole database, e.g., the partial results, obtained when applying the algorithm on 85.01% of the whole MIT-BIH arrhythmia Database, are 99.14% of sensitivity, 98.94% of positive predictive value and 01.92% of percentage error.     

Maximal and supremal tolerances in multiobjective linear programming

Let a multiobjective linear programming problem and any efficient solution be given. Tolerance analysis aims to compute interval tolerances for (possibly all) objective function coefficients such that the efficient solution remains efficient for any perturbation of the coefficients within the computed intervals. The known methods either yield tolerances that are not the maximal possible ones, or they consider perturbations of weights of the weighted sum scalarization only. We focus directly on perturbations of the objective function coefficients, which makes the approach independent on a scalarization technique used. In this paper, we propose a method for calculating the supremal tolerance (the maximal one need not exist). The main disadvantage of the method is the exponential running time in the worst case. Nevertheless, we show that the problem of determining the maximal/supremal tolerance is NP-hard, so an efficient (polynomial time) procedure is not likely to exist. We illustrate our approach on examples and present an application in transportation problems. Since the maximal tolerance may be small, we extend the notion to individual lower and upper tolerances for each objective function coefficient. An algorithm for computing maximal individual tolerances is proposed.     

Standard sensitivity analysis and additive tolerance approach in MOLP

We consider sensitivity analysis of the objective function coefficients in multiple objective linear programming (MOLP). We focus on the properties of the parameters set for which a given extreme solution is efficient. Moreover, we compare two approaches: the standard sensitivity analysis (changing only one coefficient) and the additive tolerance approach (changing all coefficients). We find the connections between these two approaches by giving a theorem describing the upper bound on the maximal additive tolerance.     

Fast construction of constant bound functions for sparse polynomials

A new method for the representation and computation of Bernstein coefficients of multivariate polynomials is presented. It is known that the coefficients of the Bernstein expansion of a given polynomial over a specified box of interest tightly bound the range of the polynomial over the box. The traditional approach requires that all Bernstein coefficients are computed, and their number is often very large for polynomials with moderately-many variables. The new technique detailed represents the coefficients implicitly and uses lazy evaluation so as to render the approach practical for many types of non-trivial sparse polynomials typically encountered in global optimization problems; the computational complexity becomes nearly linear with respect to the number of terms in the polynomial, instead of exponential with respect to the number of variables. These range-enclosing coefficients can be employed in a branch-and-bound framework for solving constrained global optimization problems involving polynomial functions, either as constant bounds used for box selection, or to construct affine underestimating bound functions. If such functions are used to construct relaxations for a global optimization problem, then sub-problems over boxes can be reduced to linear programming problems, which are easier to solve. Some numerical examples are presented and the software used is briefly introduced.     

Additive and multiplicative tolerance in multiobjective linear programming

We consider a multiobjective linear program. We propose a procedure for computing an additive and multiplicative (percentage) tolerance in which all the objective function coefficients may simultaneously and independently vary while preserving the efficiency of a given solution. For a nondegenerate basic solution, the procedure runs in polynomial time.     

INTERFACE PROBLEMS FOR ELLIPTIC DIFFERENTIAL EQUATIONS

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Limit of ratio of consecutive terms for general order-k linear homogeneous recurrences with constant coefficients

For complex linear homogeneous recursive sequences with constant coefficients we find a necessary and sufficient condition for the existence of the limit of the ratio of consecutive terms. The result can be applied even if the characteristic polynomial has not necessarily roots with modulus pairwise distinct, as in the celebrated Poincaré’s theorem. In case of existence, we characterize the limit as a particular root of the characteristic polynomial, which depends on the initial conditions and that is not necessarily the unique root with maximum modulus and multiplicity. The result extends to a quite general context the way used to find the Golden mean as limit of ratio of consecutive terms of the classical Fibonacci sequence.     

Strong asymptotics of the recurrence coefficients of orthogonal polynomials associated to the generalized Jacobi weight

We study asymptotics of the recurrence coefficients of orthogonal polynomials associated to the generalized Jacobi weight, which is a weight function with a finite number of algebraic singularities on [−1,1]. The recurrence coefficients can be written in terms of the solution of the corresponding Riemann–Hilbert (RH) problem for orthogonal polynomials. Using the steepest descent method of Deift and Zhou, we analyze the RH problem, and obtain complete asymptotic expansions of the recurrence coefficients. We will determine explicitly the order 1/n terms in the expansions. A critical step in the analysis of the RH problem will be the local analysis around the algebraic singularities, for which we use Bessel functions of appropriate order. In addition, the RH approach gives us also strong asymptotics of the orthogonal polynomials near the algebraic singularities in terms of Bessel functions.     

On accuracy,robustness and tolerances in vector Boolean optimization

A Boolean programming problem with a finite number of alternatives where initial coefficients (costs) of linear payoff functions are subject to perturbations is considered. We define robust solution as a feasible solution which for a given set of realizations of uncertain parameters guarantees the minimum value of the worst-case relative regret among all feasible solutions. For the Pareto optimality principle, an appropriate definition of the worst-case relative regret is specified. It is shown that this definition is closely related to the concept of accuracy function being recently intensively studied in the literature. We also present the concept of robustness tolerances of a single cost vector. The tolerance is defined as the maximum level of perturbation of the cost vector which does not destroy the solution robustness. We present formulae allowing the calculation of the robustness tolerance obtained for some initial costs. The results are illustrated with several numerical examples.     

Correlation and the time interval in multiple regression models

In this paper we investigate the time interval effect of multiple regression models in which some of the variables are additive and some are multiplicative. The effect on the partial regression and correlation coefficients is influenced by the selected time interval. We find that the partial regression and correlation coefficients between two additive variables approach one-period values as n increases. When one of the variables is multiplicative, they will approach zero in the limit. We also show that the decreasing speed of the n-period correlation coefficients between both multiplicative variables is faster than others, except that a one-period correlation has a higher positive value. The results of this paper can be widely applied in various fields where regression or correlation analyses are employed.     

A categorical approach to the maximum theorem

Berge's maximum theorem gives conditions ensuring the continuity of an optimised function as a parameter changes. In this paper we state and prove the maximum theorem in terms of the theory of monoidal topology and the theory of double categories.This approach allows us to generalise (the main assertion of) the maximum theorem, which is classically stated for topological spaces, to pseudotopological spaces and pretopological spaces, as well as to closure spaces, approach spaces and probabilistic approach spaces, amongst others. As a part of this we prove a generalisation of the extreme value theorem.     

An approximate lower tolerance bound for the three-parameter Weibull applied to lumber property characterization

We investigate a large sample approach for obtaining tolerance bounds where the underlying population is a three-parameter Weibull distribution. Accurate tolerance bounds could play an important role in the development of lumber standards. Properties of the maximum likelihood based approach are compared with those of the standard nonparametric tolerance procedure. The asymptotic normal approximation to the tolerance bound was found to be inadequate for most of the cases considered.     

Numerical simulation of the soft contact dynamics of an impacting bilinear oscillator

Systems constituted by moving components that make intermittent contacts with each other can be modelled by a system of ordinary differential equations containing piecewise linear terms. We consider a soft impact bilinear oscillator for which we obtain bifurcation diagrams, Lyapunov coefficients, return maps and phase portraits of the response. Besides Lyapunov coefficients diagrams, bifurcation diagrams are represented in terms of both non-dimensional time instants of contact (when the mass impacts the obstacle) and of portions of contact duration (the percentage-time interval when the material point is inside the obstacle) vs. non-dimensional external force frequency (or amplitude). The second kind of diagrams is needed because the contact duration (or the complementary flight time duration) are quantities that can easily be measured in an experiment aiming at confirming the validity of the present model. Lyapunov coefficients are evaluated converting the piecewise linear system of ordinary differential equations into a map, the so-called impact map, where time and velocity corresponding to a given impact are evaluated as functions of time and velocity corresponding to the previous impact. Thus, the usual methods related to this last map are used. The trajectories are represented in terms of return maps (all points in the time-velocity plane involved in the impact events) and phase portraits (the trajectory-itself in the displacement-velocity plane). In the bifurcation diagrams, transition between different responses is evidenced and a perfect correlation between chaotic (periodic) attractors and positive (negative) values of the maximum Lyapunov coefficient is found.     

Multioperator representation of composite compact schemes

The multioperator approach is used to obtain high-order accurate compact differences. These differences are developed to describe convective terms of differential equations, as well as mixed derivatives, source terms, and the coefficients of metric derivatives of coordinate transformations. The same principles are used to obtain high-order compact differences for representing diffusion terms. These differences underlie multioperator composite compact schemes, which are used to compute the flow past an airfoil by integrating the nonstationary Navier-Stokes equations supplemented with the equations of a turbulent viscosity model.