Interval linear constraint solving in constraint logic programming.

by Chong-kan Chiu.Thesis (M.Phil.)--Chinese University of Hong Kong, 1994.Includes bibliographical references (leaves 97-103).Chapter 1 --- Introduction --- p.1Chapter 1.1 --- Related Work --- p.2Chapter 1.2 --- Organizations of the Dissertation --- p.4Chapter 1.3 --- Notations --- p.4Chapter 2 --- Overview of ICLP(R) --- p.6Chapter 2.1 --- Basics of Interval Arithmetic --- p.6Chapter 2.2 --- Relational Interval Arithmetic --- p.8Chapter 2.2.1 --- Interval Reduction --- p.8Chapter 2.2.2 --- Arithmetic Primitives --- p.10Chapter 2.2.3 --- Interval Narrowing and Interval Splitting --- p.13Chapter 2.3 --- Syntax and Semantics --- p.16Chapter 3 --- Limitations of Interval Narrowing --- p.18Chapter 3.1 --- Computation Inefficiency --- p.18Chapter 3.2 --- Inability to Detect Inconsistency --- p.23Chapter 3.3 --- The Newton Language --- p.27Chapter 4 --- Design of CIAL --- p.30Chapter 4.1 --- The CIAL Architecture --- p.30Chapter 4.2 --- The Inference Engine --- p.31Chapter 4.2.1 --- Interval Variables --- p.31Chapter 4.2.2 --- Extended Unification Algorithm --- p.33Chapter 4.3 --- The Solver Interface and Constraint Decomposition --- p.34Chapter 4.4 --- The Linear and the Non-linear Solvers --- p.37Chapter 5 --- The Linear Solver --- p.40Chapter 5.1 --- An Interval Gaussian Elimination Solver --- p.41Chapter 5.1.1 --- Naive Interval Gaussian Elimination --- p.41Chapter 5.1.2 --- Generalized Interval Gaussian Elimination --- p.43Chapter 5.1.3 --- Incrementality of Generalized Gaussian Elimination --- p.47Chapter 5.1.4 --- Solvers Interaction --- p.50Chapter 5.2 --- An Interval Gauss-Seidel Solver --- p.52Chapter 5.2.1 --- Interval Gauss-Seidel Method --- p.52Chapter 5.2.2 --- Preconditioning --- p.55Chapter 5.2.3 --- Increment ality of Preconditioned Gauss-Seidel Method --- p.58Chapter 5.2.4 --- Solver Interaction --- p.71Chapter 5.3 --- Comparisons --- p.72Chapter 5.3.1 --- Time Complexity --- p.72Chapter 5.3.2 --- Storage Complexity --- p.73Chapter 5.3.3 --- Others --- p.74Chapter 6 --- Benchmarkings --- p.76Chapter 6.1 --- Mortgage --- p.78Chapter 6.2 --- Simple Linear Simultaneous Equations --- p.79Chapter 6.3 --- Analysis of DC Circuit --- p.80Chapter 6.4 --- Inconsistent Simultaneous Equations --- p.82Chapter 6.5 --- Collision Problem --- p.82Chapter 6.6 --- Wilkinson Polynomial --- p.85Chapter 6.7 --- Summary and Discussion --- p.86Chapter 6.8 --- Large System of Simultaneous Equations --- p.87Chapter 6.9 --- Comparisons Between the Incremental and the Non-Incremental Preconditioning --- p.89Chapter 7 --- Concluding Remarks --- p.93Chapter 7.1 --- Summary and Contributions --- p.93Chapter 7.2 --- Future Work --- p.95Bibliography --- p.9

Value constraints in the CLP scheme

This paper addresses the question of how to incorporate constraint propagation into logic programming. A likely candidate is the CLP scheme, which allows one to exploit algorithmic opportunities while staying within logic programming semantics. CLP($cal R$) is an example: it combines logic programming with the algorithms for solving linear equalities and inequalities. In this paper we describe two contrasting constraint store management strategies within the CLP scheme. One leads to CLP($cal R$), while the other, which we call value constraints, supports consistency methods such as arc consistency and interval constraints. In value constraints, the infer step of the CLP scheme is the application of a consistency operator acting on the active constraints. We show how the continued application of the infer step can be optimized and that such optimization is equivalent to the Waltz algorithm for constraint propagation. Using the Lassez-Maher fixpoint theory of chaotic iterations, we show that the Waltz algorithm does not necessarily converge to a fixpoint, but that it does so in the finitary case

Value constraints in the CLP scheme

This paper addresses the question of how to incorporate constraint propagation into logic programming. A likely candidate is the CLP scheme, which allows one to exploit algorithmic opportunities while staying within logic programming semantics. CLP($cal R$) is an example: it combines logic programming with the algorithms for solving linear equalities and inequalities. In this paper we describe two contrasting constraint store management strategies within the CLP scheme. One leads to CLP($cal R$), while the other, which we call value constraints, supports consistency methods such as arc consistency and interval constraints. In value constraints, the infer step of the CLP scheme is the application of a consistency operator acting on the active constraints. We show how the continued application of the infer step can be optimized and that such optimization is equivalent to the Waltz algorithm for constraint propagation. Using the Lassez-Maher fixpoint theory of chaotic iterations, we show that the Waltz algorithm does not necessarily converge to a fixpoint, but that it does so in the finitary case

Inner Regions and Interval Linearizations for Global Optimization

International audienceResearchers from interval analysis and constraint (logic) programming communities have studied intervals for their ability to manage infinite solution sets of numerical constraint systems. In particular, inner regions represent subsets of the search space in which all points are solutions. Our main contribution is the use of recent and new inner region extraction algorithms in the upper bounding phase of constrained global optimization. Convexification is a major key for efficiently lower bounding the objective function. We have adapted the convex interval taylorization proposed by Lin & Stadtherr for producing a reliable outer and inner polyhedral approximation of the solution set and a linearization of the objective function. Other original ingredients are part of our optimizer, including an efficient interval constraint propagation algorithm exploiting monotonicity of functions. We end up with a new framework for reliable continuous constrained global optimization. Our interval B&B is implemented in the interval-based explorer Ibex and extends this free C++ library. Our strategy significantly outperforms the best reliable global optimizers

Optimal External Memory Interval Management

This is the published version. Copyright © 2003 Society for Industrial and Applied MathematicsIn this paper we present the external interval tree, an optimal external memory data structure for answering stabbing queries on a set of dynamically maintained intervals. The external interval tree can be used in an optimal solution to the dynamic interval management problem, which is a central problem for object-oriented and temporal databases and for constraint logic programming. Part of the structure uses a weight-balancing technique for efficient worst-case manipulation of balanced trees, which is of independent interest. The external interval tree, as well as our new balancing technique, have recently been used to develop several efficient external data structures

Modular Constraint Solver Cooperation via Abstract Interpretation

Cooperation among constraint solvers is difficult because different solving paradigms have different theoretical foundations. Recent works have shown that abstract interpretation can provide a unifying theory for various constraint solvers. In particular, it relies on abstract domains which capture constraint languages as ordered structures. The key insight of this paper is viewing cooperation schemes as abstract domains combinations. We propose a modular framework in which solvers and cooperation schemes can be seamlessly added and combined. This differs from existing approaches such as SMT where the cooperation scheme is usually fixed (e.g., Nelson-Oppen). We contribute to two new cooperation schemes: (i) interval propagators completion that allows abstract domains to exchange bound constraints, and (ii) delayed product which exchanges over-approximations of constraints between two abstract domains. Moreover, the delayed product is based on delayed goal of logic programming, and it shows that abstract domains can also capture control aspects of constraint solving. Finally, to achieve modularity, we propose the shared product to combine abstract domains and cooperation schemes. Our approach has been fully implemented, and we provide various examples on the flexible job shop scheduling problem. Under consideration for acceptance in TPLP.Comment: Paper presented at the 36th International Conference on Logic Programming (ICLP 2020), University Of Calabria, Rende (CS), Italy, September 2020, 17 pages. v2: Fix an example in Section 3.2 (improved closure

Optimal External Memory Interval Management

This is the publisher's version, which is being shared on KU Scholarworks with permission. The original version may be found at the following link: http://dx.doi.org/10.1137/S009753970240481XIn this paper we present the external interval tree, an optimal external memory data structure for answering stabbing queries on a set of dynamically maintained intervals. The external interval tree can be usedin an optimal solution to the dynamic interval management problem, which is a central problem for object-orientedandtemp oral databases andfor constraint logic programming. Part of the structure uses a weight-balancing technique for efficient worst-case manipulation of balanced trees, which is of independent interest. The external interval tree, as well as our new balancing technique, have recently been used to develop several efficient external data structures

Interval propagation and search on directed acyclic graphs for numerical constraint solving

The fundamentals of interval analysis on directed acyclic graphs (DAGs) for global optimization and constraint propagation have recently been proposed in Schichl and Neumaier (J. Global Optim. 33, 541-562, 2005). For representing numerical problems, the authors use DAGs whose nodes are subexpressions and whose directed edges are computational flows. Compared to tree-based representations [Benhamou etal. Proceedings of the International Conference on Logic Programming (ICLP'99), pp. 230-244. Las Cruces, USA (1999)], DAGs offer the essential advantage of more accurately handling the influence of subexpressions shared by several constraints on the overall system during propagation. In this paper we show how interval constraint propagation and search on DAGs can be made practical and efficient by: (1) flexibly choosing the nodes on which propagations must be performed, and (2) working with partial subgraphs of the initial DAG rather than with the entire graph. We propose a new interval constraint propagation technique which exploits the influence of subexpressions on all the constraints together rather than on individual constraints. We then show how the new propagation technique can be integrated into branch-and-prune search to solve numerical constraint satisfaction problems. This algorithm is able to outperform its obvious contenders, as shown by the experiment