Interval Arithmetic Techniques Research Articles

Worst-Case Optimal Covering of Rectangles by Disks

We provide the solution for a fundamental problem of geometric optimization by giving a complete characterization of worst-case optimal disk coverings of rectangles: For any λ≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda \\ge 1$$\\end{document}, the critical covering area A∗(λ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A^*(\\lambda )$$\\end{document} is the minimum value for which any set of disks with total area at least A∗(λ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A^*(\\lambda )$$\\end{document} can cover a rectangle of dimensions λ×1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda \ imes 1$$\\end{document}. We show that there is a threshold value λ2=7/2-1/4≈1.035797…\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda _2 = \\sqrt{\\sqrt{7}/2 - 1/4} \\approx 1.035797\\ldots $$\\end{document}, such that for λ<λ2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda <\\lambda _2$$\\end{document} the critical covering area A∗(λ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A^*(\\lambda )$$\\end{document} is A∗(λ)=3πλ216+532+9256λ2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A^*(\\lambda )=3\\pi \\left( \\frac{\\lambda ^2}{16} +\\frac{5}{32} + \\frac{9}{256\\lambda ^2}\\right) $$\\end{document}, and for λ≥λ2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda \\ge \\lambda _2$$\\end{document}, the critical area is A∗(λ)=π(λ2+2)/4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A^*(\\lambda )=\\pi (\\lambda ^2+2)/4$$\\end{document}; these values are tight. For the special case λ=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda =1$$\\end{document}, i.e., for covering a unit square, the critical covering area is 195π256≈2.39301…\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\frac{195\\pi }{256}\\approx 2.39301\\ldots $$\\end{document}. The proof uses a careful combination of manual and automatic analysis, demonstrating the power of the employed interval arithmetic technique.

Computing Enclosures for the Matrix Exponential

We present a review of old interval arithmetic techniques, and develop new ones, for computing enclosures for all entries of the exact exponential of a matrix. This means that all the rounding and ...

Rigorous packing of unit squares into a circle

This paper considers the task of finding the smallest circle into which one can pack a fixed number of non-overlapping unit squares that are free to rotate. Due to the rotation angles, the packing of unit squares into a container is considerably harder to solve than their circle packing counterparts. Therefore, optimal arrangements were so far proved to be optimal only for one or two unit squares. By a computer-assisted method based on interval arithmetic techniques, we solve the case of three squares and find rigorous enclosures for every optimal arrangement of this problem. We model the relation between the squares and the circle as a constraint satisfaction problem (CSP) and found every box that may contain a solution inside a given upper bound of the radius. Due to symmetries in the search domain, general purpose interval methods are far too slow to solve the CSP directly. To overcome this difficulty, we split the problem into a set of subproblems by systematically adding constraints to the center of each square. Our proof requires the solution of 6, 43 and 12 subproblems with 1, 2 and 3 unit squares respectively. In principle, the method proposed in this paper generalizes to any number of squares.

Numerical validation of blow-up solutions of ordinary differential equations

This paper focuses on blow-up solutions of ordinary differential equations (ODEs). We present a method for validating blow-up solutions and their blow-up times, which is based on compactifications and the Lyapunov function validation method. The necessary criteria for this construction can be verified using interval arithmetic techniques. Some numerical examples are presented to demonstrate the applicability of our method.

Evolutionary technique based goal programming approach to chance constrained interval valued bilevel programming problems

The real world multiobjective decision environment involves great complexity and uncertainity. Many decision making problems often need to be modelled as a class of bilevel programming problems with inexact coefficients and chance constraints. To deal with these problems, a genetic algorithm (GA) based goal programming (GP) procedure for solving interval valued bilevel programming (BLP) problems in a large hierarchical decision making and planning organization is proposed. In the model formulation of the problem the chance constraints are converted to their deterministic equivalent using the notion of mean and variance. Further, the individual best and least solutions of the objectives of the decision makers (DMs) located at different hierarchical levels are determined by using GA method. The target intervals for achievement of each of the objectives as well as the target interval of the decision vector controlled by the upper-level DM are defined. Then, using interval arithmetic technique the interval valued objectives and control vectors are transformed into the conventional form of goal by introducing under- and over-deviational variables to each of them. In the solution process, both the aspects of minsum and minmax GP formulations are adopted to minimize the lower bounds of the regret intervals for goal achievement within the specified interval from the optimistic point of view. The potential use of the approach is illustrated by a numerical example

A blind digital image watermarking method based on the dyadic wavelet packet transform and fast interval arithmetic techniques

We propose a new digital image watermarking method based on the dyadic wavelet packet transform (DYWPT) and fast interval arithmetic (IA) techniques. Because the DYWPT has a redundant representation, like the dyadic wavelet transform (DYWT), the amount of information that the watermark must contain is greater than in the case of methods based on ordinary discrete wavelet transforms (DWTs) and the discrete wavelet packet transform (DWPT). However, the order of the high frequency components is not necessarily the same as the order of their frequency domain. Therefore, in our approach, we rearrange the order of the high frequency components in descending order of frequency components and embed a watermark selectively into higher frequency components. Our watermark is a ternary-valued logo that is embedded into higher frequency components through use of the DYWPT and fast IA techniques. We describe our watermarking procedure in detail and present experimental results demonstrating that our method produces watermarked images that have better quality and are robust with respect to various types of attacks, including marking, clipping, median filtering, contrast tuning (histeq and imadjust commands in the MATLAB Image Processing Toolbox), addition of Gaussian white noise, addition of salt &amp; pepper noise, JPEG and JPEG 2000 compressions, rotation, and resizing.

Solving capacitor placement problem considering uncertainty in load variation

The paper reports on the solution of the capacitor placement problem in distribution system considering uncertainty in the variation of loads. Solution techniques available in the literature generally consider load variation as deterministic. In the present paper uncertainty in load variation is considered using fuzzy interval arithmetic technique. Load variations are represented as lower and upper bounds around base levels. Both fixed and switchable capacitors have been considered and results for standard test systems are presented.

Internal Model Control Based on Interval Arithmetic for Suspension Systems

This paper deals with the application of interval arithmetic in internal model control (IMC) for suspension systems. The IMC problem is expressed as an optimization of filter parameters. And the optimization can be solved by branch-and-bound scheme with interval arithmetic techniques. We present the algorithm for the model of suspension systems. The effectiveness of the algorithm is demonstrated through simulation results.

A priority-based goal programming method for solving academic personnel planning problems with interval-valued resource goals in university management system

This paper describes a goal programming (GP) procedure for modelling and solving academic resource planning problems in university management system. In the academic resource planning context, certain objectives having the characteristics of fractional programming problems are considered as interval-valued goals to make a satisfactory decision regarding staff allocation for smooth functioning of the academic activities of the departments. In the proposed approach, the interval-valued goals are first converted into the standard goals by using interval arithmetic technique. Then, the fractional goals are transformed into linear goals by using linearisation approach to solve the problem by employing linear GP methodology. In the model formulation of the problem, both the aspects of GP, minsum and minimax approaches, are addressed to construct the goal achievement function for minimising the possible regret towards achieving the goal values within the target intervals specified by the decision maker (DM) in the decision making environment. A demonstrative case example of the University of Kalyani, West Bengal (W.B.), India is considered to expound the proposed model.

A volume flexible economic production lot-sizing problem with imperfect quality and random machine failure in fuzzy-stochastic environment

An “economic production lot size” (EPLS) model for an item with imperfect quality is developed by considering random machine failure. Breakdown of the manufacturing machines is taken into account by considering its failure rate to be random (continuous). The production rate is treated as a decision variable. It is assumed that some defective units are produced during the production process. Machine breakdown resulting in idle time of the respective machine which leads to additional cost for loss of manpower is taken into account. It is assumed that the production of the imperfect quality units is a random variable and all these units are treated as scrap items that are completely wasted. The models have been formulated as profit maximization problems in stochastic and fuzzy-stochastic environments by considering some inventory parameters as imprecise in nature. In a fuzzy-stochastic environment, using interval arithmetic technique, the interval objective function has been transformed into an equivalent deterministic multi-objective problem. Finally, multi-objective problem is solved by Global Criteria Method (GCM). Stochastic and fuzzy-stochastic problems and their significant features are illustrated by numerical examples. Using the result of the stochastic model, sensitivity of the nearer optimal solution due to changes of some key parameters are analysed.

Interval arithmetic techniques for the design of controllers for nonlinear dynamical systems with applications in mechatronics. II

In this paper, we give an overview of interval arithmetic techniques for both the offline and online verification of robust control and observer strategies. In Part 1 of this paper, we have introduced basic interval arithmetic techniques focusing on offline applications while we deal with their online application in this Part 2. In contrast to the offline tasks considered in Part 1, in which guaranteed enclosures of all admissible solutions to a specific control problem are computed, real-time requirements have to be considered in online tasks. This means that we have to ensure to find a guaranteed admissible solution within a limited computing time. Therefore, the adapted goal in the online design of controllers and observers must be to determine at least one solution, and not necessarily all, such that feasibility under the influence of uncertainties is proven. After a summary of the basic computational procedures, first experimental results for the real-time application of interval tools for feedforward control and disturbance estimation are presented for the temperature control of a distributed heating system.

Interval arithmetic techniques for the design of controllers for nonlinear dynamical systems with applications in mechatronics

Interval arithmetic techniques for the design of controllers for nonlinear dynamical systems with applications in mechatronics

An Affine Arithmetic-Based Methodology for Reliable Power Flow Analysis in the Presence of Data Uncertainty

Power flow studies are typically used to determine the steady state or operating conditions of power systems for specified sets of load and generation values, and is one of the most intensely used tools in power engineering. When the input conditions are uncertain, numerous scenarios need to be analyzed to cover the required range of uncertainty. Under such conditions, reliable solution algorithms that incorporate the effect of data uncertainty into the power flow analysis are required. To address this problem, this paper proposes a new solution methodology based on the use of affine arithmetic, which is an enhanced model for self-validated numerical analysis in which the quantities of interest are represented as affine combinations of certain primitive variables representing the sources of uncertainty in the data or approximations made during the computation. The application of this technique to the power flow problem is explained in detail, and several numerical results are presented and discussed, demonstrating the effectiveness of the proposed methodology, especially in comparison to previously proposed interval arithmetic's techniques.

MSB-First Interval-Bounded Variable-Precision Real-Time Arithmetic Unit

This paper presents a paradigm of real-time processing on the lowest level of computing systems: the arithmetic unit. The arithmetic unit based on this principle containing addition, subtraction, multiplication and division operations is described. The development of the computation model is based on the Soft Computing and the Imprecise Computation paradigms, combined with the MSB- First and the Interval Arithmetic techniques. Those paradigms and techniques give the arithmetic unit design the ability to compute with precisions as a function of time available or accuracy needed. The predictability of processing time and result's accuracy are obtained by means of processing granularity of k- bits and by using look-up tables. We present an evaluation of the operation in time delay and computation accuracy that shows significant performance improvement over conventional arithmetic unit architecture, that is, the ability to produce intermediate-result during execution time, to give certainty in computation accuracy even before the process finish time by providing two intermediate-results, which act as the lower and upper bound of the real and complete computation result, and finally, gain high computation accuracy from the early time of the execution process.

The Adaptive Convexification Algorithm: A Feasible Point Method for Semi-Infinite Programming

We present a new numerical solution method for semi-infinite optimization problems. Its main idea is to adaptively construct convex relaxations of the lower level problem, replace the relaxed lower level problems equivalently by their Karush–Kuhn–Tucker conditions, and solve the resulting mathematical programs with complementarity constraints. This approximation produces feasible iterates for the original problem. The convex relaxations are constructed with ideas from the $\alpha$BB method of global optimization. The necessary upper bounds for second derivatives of functions on box domains can be determined using the techniques of interval arithmetic, where our algorithm already works if only one such bound is available for the problem. We show convergence of stationary points of the approximating problems to a stationary point of the original semi-infinite problem within arbitrarily given tolerances. Numerical examples from Chebyshev approximation and design centering illustrate the performance of the method.

A deteriorating multi-item inventory model with fuzzy costs and resources based on two different defuzzification techniques

Normally inventory models of deteriorating items, such as food products, vegetables, etc. involve imprecise parameters, like imprecise inventory costs, fuzzy storage area, fuzzy budget allocation, etc. In this paper, we aim to provide two defuzzification techniques for two fuzzy inventory models using (i) extension principle and duality theory of non-linear programming and (ii) interval arithmetic. On the basis of Zadeh’s extension principle, two non-linear programs parameterized by the possibility level α are formulated to calculate the lower and upper bounds of the minimum average cost at α -level, through which the membership function of the objective function is constructed. In interval arithmetic technique the interval objective function has been transformed into an equivalent deterministic multi-objective problem defined by the left and right limits of the interval. This formulation corresponds to the possibility level, α = 0.5. Finally, the multi-objective problem is solved by a multi-objective genetic algorithm (MOGA). The model has been illustrated through a numerical example and solved for different values of possibility level, α through extension principle and for α = 0.5 via MOGA. As a particular case, the results have been obtained for the inventory model without deterioration. Results from two methods for α = 0.5 are compared.

Reduction of overestimation in interval arithmetic simulation of biological wastewater treatment processes

A novel interval arithmetic simulation approach is introduced in order to evaluate the performance of biological wastewater treatment processes. Such processes are typically modeled as dynamical systems where the reaction kinetics appears as additive nonlinearity in state. In the calculation of guaranteed bounds of state variables uncertain parameters and uncertain initial conditions are considered. The recursive evaluation of such systems of nonlinear state equations yields overestimation of the state variables that is accumulating over the simulation time. To cope with this wrapping effect, innovative splitting and merging criteria based on a recursive uncertain linear transformation of the state variables are discussed. Additionally, re-approximation strategies for regions in the state space calculated by interval arithmetic techniques using disjoint subintervals improve the simulation quality significantly if these regions are described by several overlapping subintervals. This simulation approach is used to find a practical compromise between computational effort and simulation quality. It is pointed out how these splitting and merging algorithms can be combined with other methods that aim at the reduction of overestimation by applying consistency techniques. Simulation results are presented for a simplified reduced-order model of the reduction of organic matter in the activated sludge process of biological wastewater treatment.

Using an interval branch‐and‐bound algorithm in the Hartree–Fock method

AbstractThe Hartree–Fock (HF) method is widely used to obtain atomic and molecular electronic wave functions, based on the minimization of a functional of the energy. We propose to use a deterministic global optimization algorithm, based on a branch‐and‐bound method, that applies techniques of interval arithmetic. This algorithm is applied directly to the minimization of the energy expression derived from the HF method. The proposed approach was successfully applied to the ground state of He and Be atoms. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005

A deterministic approach for global minimization of molecular potential energy functions

AbstractThe problem of minimizing a molecular potential energy function is an example of a global optimization problem. Computing the global minimum of this function is very difficult, because it typically has a very large number of local minima that grows exponentially with the problem size. Most of the methods developed to resolve this problem are stochastic or heuristic methods. In this article we use a deterministic algorithm based on a branch‐and‐bound method that employs techniques of interval arithmetic. Using this algorithm, we can guarantee that the actual global minimum is found. The proposed approach was successfully applied to a set of test problems containing up to 22 degrees of freedom. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem 95: 336–343, 2003

Radial distribution system power flow using interval arithmetic

This paper reports on the application of interval arithmetic technique for balanced radial distribution system power flow analysis. Interval arithmetic takes care of the uncertainty in the input parameters and provides strict bounds for the solution of the problem. In this paper, uncertainties only in the input load parameters are considered. The results are compared with the results obtained from repeated load-flow simulations.