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42
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 12 (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.
Inclusion Isotone Extended Interval Arithmetic  A Toolbox Update
, 1996
"... In this report we deal with the correct formulation of a special extended interval arithmetic in the context of interval Newton like methods. We first demonstrate some of the problems arising from selected older definitions. Then we investigate the basic aim, concept, and properties important for de ..."
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Cited by 6 (0 self)
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In this report we deal with the correct formulation of a special extended interval arithmetic in the context of interval Newton like methods. We first demonstrate some of the problems arising from selected older definitions. Then we investigate the basic aim, concept, and properties important for defining a correct extended interval division. Finally, we give a proper way for defining the extended interval operations needed in our special context, and we prove their inclusion isotonicity. Additionally, we give some sample applications. We conclude with two updated implementations of our extended interval operations in the toolbox environments [2] and [3].
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 "size" o ..."
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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...
Algebraic Solutions to a Class of Interval Equations
 J. UNIVERSAL COMPUTER SCIENCE
, 1998
"... The arithmetic on the extended set of proper and improper intervals is an algebraic completion of the conventional interval arithmetic and thus facilitates the explicit solution of certain interval algebraic problems. Due to the existence of inverse elements with respect to addition and multiplicati ..."
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Cited by 4 (4 self)
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The arithmetic on the extended set of proper and improper intervals is an algebraic completion of the conventional interval arithmetic and thus facilitates the explicit solution of certain interval algebraic problems. Due to the existence of inverse elements with respect to addition and multiplication operations certain interval algebraic equations can be solved by elementary algebraic transformations. The conditionally distributive relations between extended intervals allow that complicated interval algebraic equations, multiincident on the unknown variable, be reduced to simpler ones. In this paper we give the general type of "pseudolinear" interval equations in the extended interval arithmetic. The algebraic solutions to a pseudolinear interval equation in one variable are studied. All numeric and parametric algebraic solutions, as well as the conditions for nonexistence of the algebraic solution to some basic types pseudolinear interval equations in one variable are found. Some examples leading to algebraic solution of the equations under consideration and the extra functionalities for performing true symbolicalgebraic manipulations on interval formulae in a Mathematica package are discussed.
Directed Interval Arithmetic in Mathematica: Implementation and Applications
, 1996
"... This report presents an experimental Mathematica ..."