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59
Universally Quantified Interval Constraints
 PROCEEDINGS OF THE 6TH INTERNATIONAL CONFERENCE ON PRINCIPLES AND PRACTICE OF CONSTRAINT PROGRAMMING
, 2000
"... Nonlinear real constraint systems with universally and/or existentially quantified variables often need be solved in such contexts as control design or sensor planning. To date, these systems are mostly handled by computing a quantifierfree equivalent form by means of Cylindrical Algebraic Decompo ..."
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Cited by 46 (0 self)
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Nonlinear real constraint systems with universally and/or existentially quantified variables often need be solved in such contexts as control design or sensor planning. To date, these systems are mostly handled by computing a quantifierfree equivalent form by means of Cylindrical Algebraic Decomposition (CAD). However, CAD restricts its input to be conjunctions and disjunctions of polynomial constraints with rational coefficients, while some applications such as camera control involve systems with arbitrary forms where time is the only universally quantified variable. In this paper, the handling of universally quantified variables is first related to the computation of innerapproximation of real relations.
On directed interval arithmetic and its applications
, 1995
"... We discuss two closely related interval arithmetic systems: i) the system of directed (generalized) intervals studied by E. Kaucher, and ii) the system of normal intervals together with the outer and inner interval operations. A relation between the two systems becomes feasible due to introduction ..."
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Cited by 16 (4 self)
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We discuss two closely related interval arithmetic systems: i) the system of directed (generalized) intervals studied by E. Kaucher, and ii) the system of normal intervals together with the outer and inner interval operations. A relation between the two systems becomes feasible due to introduction of special notations and a socalled normal form of directed intervals. As an application, it has been shown that both interval systems can be used for the computation of tight inner and outer inclusions of ranges of functions and consequently for the development of software for automatic computation of ranges of functions.
Standardized notation in interval analysis
 In Proc. XIII Baikal International Schoolseminar “Optimization methods and their applications”. Vol. 4 “Interval analysis”. Irkutsk: Institute of Energy Systems SB RAS
, 2002
"... A standard for the notation of the most used quantities and operators in interval analysis is proposed. 1 1 ..."
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Cited by 13 (2 self)
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A standard for the notation of the most used quantities and operators in interval analysis is proposed. 1 1
Idempotent Interval Analysis and Optimization Problems
 RELIABLE COMPUTING
, 2001
"... Many problems in optimization theory are strongly nonlinear in the traditional sense but possess a hidden linear structure over suitable idempotent semirings. After an overview of ‘Idempotent Mathematics ’ with an emphasis on matrix theory, interval analysis over idempotent semirings is developed. ..."
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Cited by 11 (1 self)
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Many problems in optimization theory are strongly nonlinear in the traditional sense but possess a hidden linear structure over suitable idempotent semirings. After an overview of ‘Idempotent Mathematics ’ with an emphasis on matrix theory, interval analysis over idempotent semirings is developed. The theory is applied to construction of exact interval solutions to the interval discrete stationary Bellman equation. Solution of an interval system is typically NPhard in the traditional interval linear algebra; in the idempotent case it is polynomial. A generalization to the case of positive semirings is outlined.
Multiplication distributivity of proper and improper intervals
 RELIABLE COMPUTING
, 2001
"... The arithmetic on an extended set of proper and improper intervals presents algebraic completion of the conventional interval arithmetic allowing thus efficient solution of some interval algebraic problems. In this paper we summarize and present all distributive relations, known by now, on multiplic ..."
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Cited by 10 (0 self)
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The arithmetic on an extended set of proper and improper intervals presents algebraic completion of the conventional interval arithmetic allowing thus efficient solution of some interval algebraic problems. In this paper we summarize and present all distributive relations, known by now, on multiplication and addition of generalized (proper and improper) intervals.
Solving interval constraints by linearization in computeraided design. Reliable Computing
, 2006
"... Abstract. Current parametric CAD systems require geometric parameters to have fixed values. Specifying fixed parameter values implicitly adds rigid constraints on the geometry, which have the potential to introduce conflicts during the design process. This paper presents a soft constraint representa ..."
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Cited by 6 (5 self)
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Abstract. Current parametric CAD systems require geometric parameters to have fixed values. Specifying fixed parameter values implicitly adds rigid constraints on the geometry, which have the potential to introduce conflicts during the design process. This paper presents a soft constraint representation scheme based on nominal interval. Interval geometric parameters capture inexactness of conceptual and embodiment design, uncertainty in detail design, as well as boundary information for design optimization. To accommodate underconstrained and overconstrained design problems, a doubleloop GaussSeidel method is developed to solve linear constraints. A symbolic preconditioning procedure transforms nonlinear equations to separable form. Inequalities are also transformed and integrated with equalities. Nonlinear constraints can be bounded by piecewise linear enclosures and solved by linear methods iteratively. A sensitivity analysis method that differentiates active and inactive constraints is presented for design refinement. 1.
Simplification of SymbolicNumerical Interval Expressions
 in Gloor, O. (Ed.): Proceedings of the 1998 International Symposium on Symbolic and Algebraic Computation, ACM
, 1998
"... Although interval arithmetic is increasingly used in combination with computer algebra and other methods, both approaches  symbolicalgebraic and intervalarithmetic  are used separately. Implementing symbolic interval arithmetic seems not suitable due to the exponential growth in the "si ..."
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Cited by 5 (2 self)
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Although interval arithmetic is increasingly used in combination with computer algebra and other methods, both approaches  symbolicalgebraic and intervalarithmetic  are used separately. Implementing symbolic interval arithmetic seems not suitable due to the exponential growth in the "size" of the endpoints. In this paper we propose a methodology for "true" symbolicalgebraic manipulations on symbolicnumerical interval expressions involving interval variables instead of symbolic intervals. Due to the better algebraic properties, resembling to classical analysis, and the containment of classical interval arithmetic as a special case, we consider the algebraic extension of conventional interval arithmetic as an appropriate environment for solving interval algebraic problems. Based on the distributivity relations, a general framework for simplification of symbolicnumerical expressions involving intervals is given and some of the wider implications of the theory pertaining to inte...
Diagrammatic representation for interval arithmetic
 LINEAR ALGEBRA AND ITS APPLICATIONS
, 2001
"... The paper presents a diagrammatic representation of a standard interval space (the socalled MRdiagram), and shows how to represent and perform interval arithmetic and derive its various properties using the diagram. The representation is an extension and refinement of the ISdiagram representation ..."
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Cited by 4 (0 self)
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The paper presents a diagrammatic representation of a standard interval space (the socalled MRdiagram), and shows how to represent and perform interval arithmetic and derive its various properties using the diagram. The representation is an extension and refinement of the ISdiagram representation devised earlier by the author to represent interval relations. First, the MRdiagram is defined together with appropriate graphical notions and constructions for basic interval relations and operations. Second, diagrammatic constructions for all standard arithmetic operations are presented. Several examples of the use of these constructions to aid reasoning about various simple, though nontrivial, properties of interval arithmetic are included in order to show how the representation facilitates both deeper understanding of the subject matter and reasoning about its properties.
Imprecise probabilities with a generalized interval form
 Proc. 3rd Int. Workshop on Reliability Engineering Computing (REC'08
, 2008
"... Abstract. Di erent representations of imprecise probabilities have been proposed, such as behavioral theory, evidence theory, possibility theory, probability bound analysis, Fprobabilities, fuzzy probabilities, and clouds. These methods use intervalvalued parameters to discribe probability distrib ..."
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Cited by 4 (3 self)
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Abstract. Di erent representations of imprecise probabilities have been proposed, such as behavioral theory, evidence theory, possibility theory, probability bound analysis, Fprobabilities, fuzzy probabilities, and clouds. These methods use intervalvalued parameters to discribe probability distributions such that uncertainty is distinguished from variability. In this paper, we proposed a new form of imprecise probabilities based on generalized or modal intervals. Generalized intervals are algebraically closed under Kaucher arithmetic, which provides a concise representation and calculus structure as an extension of precise probabilities. With the separation between proper and improper interval probabilities, focal and nonfocal events are di erentiated based on the modalities and logical semantics of generalized interval probabilities. Focal events have the semantics of critical, uncontrollable, speci ed, etc. in probabilistic analysis, whereas the corresponding nonfocal events are complementary, controllable, and derived. A generalized imprecise conditional probability is de ned based on unconditional interval probabilities such that the algebraic relation between conditional and marginal interval probabilities is maintained. A Bayes ' rule with generalized intervals (GIBR) is also proposed. The GIBR allows us to interpret the logic relationship between interval prior and posterior probabilities.