A new framework for sharp and efficient resolution of NCSP with manifolds of solutions

When numerical CSPs are used to solve systems of n equations with n variables, the preconditioned interval Newton operator plays two key roles: First it allows handling the n equations as a global constraint, hence achieving a powerful contraction. Second it can prove rigorously the existence of solutions. However, none of these advantages can be used for under-constrained systems of equations, which have manifolds of solutions. A new framework is proposed in this paper to extend the advantages of the preconditioned interval Newton to under-constrained systems of equations. This is achieved simply by allowing domains of the NCSP to be parallelepipeds, which generalize the boxes usually used as domains.     

Simultaneous automated design of structured QFT controller and prefilter using nonlinear programming

This paper describes a nonlinear programming‐based robust design methodology for controllers and prefilters of a predefined structure for the linear time‐invariant systems involved in the quantitative feedback theory. This controller and prefilter synthesis problem is formulated as a single optimization problem with a given performance optimization objective and constraints enforcing stability and various specifications usually enforced in the quantitative feedback theory. The focus is set on providing constraints expression that can be used in standard nonlinear programming solvers. The nonlinear solver then computes in a single‐step controller and prefilter design parameters that satisfy the prescribed constraints and maximizes the performance optimization objective. The effectiveness of the proposed approach is demonstrated through a variety of difficult design cases like resonant plants, open‐loop unstable plants, and plants with variation in the time delay. Copyright © 2016 John Wiley & Sons, Ltd.     

On the Combination of Interval Constraint Solvers

This paper tackles the combination of interval methods for solving nonlinear systems. A cooperative strategy of application of elementary solvers is designed in order to accelerate the whole computation while weakening the local domain contractions. It is implemented in a prototype solver which efficiently combines interval-based local consistencies and the multidimensional interval Newton method. A set of experiments shows a gain of one order of magnitude on average with respect to Numerica.     

Automatic Generation of Numerical Redundancies for Non-Linear Constraint Solving

In this paper we present a framework for the cooperation of symbolic and propagation-based numerical solvers over the real numbers. This cooperation is expressed in terms of fixed points of closure operators over a complete lattice of constraint systems. In a second part we instantiate this framework to a particular cooperation scheme, where propagation is associated to pruning operators implementing interval algorithms enclosing the possible solutions of constraint systems, whereas symbolic methods are mainly devoted to generate redundant constraints. When carefully chosen, it is well known that the addition of redundant constraint drastically improve the performances of systems based on local consistency (e.g. Prolog IV or Newton). We propose here a method which computes sets of redundant polynomials called partial Gröbner bases and show on some benchmarks the advantages of such computations.     

Horner's Rule for Interval Evaluation Revisited

Interval arithmetic can be used to enclose the range of a real function over a domain. However, due to some weak properties of interval arithmetic, a computed interval can be much larger than the exact range. This phenomenon is called dependency problem. In this paper, Horner's rule for polynomial interval evaluation is revisited. We introduce a new factorization scheme based on well-known symbolic identities in order to handle the dependency problem of interval arithmetic. The experimental results show an improvement of 25% of the width of computed intervals with respect to Horner's rule. Received December 14, 2001; revised March 27, 2002 Published online: July 8, 2002