Abstract
In this paper, we show that the problem of computing the smallest interval submatrix of a given interval matrix [A] which contains all symmetric positive semi-definite (PSD) matrices of [A], is a linear matrix inequality (LMI) problem, a convex optimization problem over the cone of positive semidefinite matrices, that can be solved in polynomial time. From a constraint viewpoint, this problem corresponds to projecting the global constraint PSD (A) over its domain [A]. Projecting such a global constraint, in a constraint propagation process, makes it possible to avoid the decomposition of the PSD constraint into primitive constraints and thus increases the efficiency and the accuracy of the resolution.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Adjiman, C., Dallwig, S., Floudas, C., and Neumaier, A.: A Global Optimization Method, αBB, for General Twice-Differentiable Constrained NLPs — I Theoretical Advances, Computers and Chemical Engineering 22 (1998), pp. 1137–1158.
Ben-Tal, A. and Nemirovskii, A.: Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications, SIAM, Philadelphia, 2001.
Bliek, C., Spellucci, P., Vincente, L., Neumaier, A., Granvilliers, L., Huens, E., Hentenryck, P. V., Sam-Haroud, D., and Faltings, B.: Algorithms for Solving Nonlinear Constrained and Optimisation Problems: State of the Art, Progress report of the COCONUT project, 2001, http://www.mat.univie.ac.at/~neum/glopt/coconut/cocbib.html.
Boyd, S. and Vanbenberghe, L.: Convex Optimization, Lecture Notes at Stanford University and University of California at Los Angeles, 2003, http://www.stanford.edu/~boyd.
Davey, B. A. and Priestley, H. A.: Introduction to Lattices and Order, Cambridge University Press, 2002.
Gahinet, P. and Nemirovskii, A.: LMI Lab: A Package for Manipulating and Solving LMIs, INRIA Rocquencourt, France, 1993.
Gahinet, P., Nemirovskii, A., Laub, A. J., and Chilali, M.: The LMI Control Toolbox for Use with Matlab, The MathWorks, Natick, 1995.
Hansen, E. R.: Global Optimization Using Interval Analysis, Marcel Dekker, New York, 1992.
Jansson, C.: A Rigorous Lower Bound for the Optimal Value of Convex Optimization Problems, Journal of Global Optimization 28 (2004), pp. 121–137.
Jaulin, L., Kieffer, M., Didrit, O., and Walter, E.: Applied Interval Analysis, with Examples in Parameter and State Estimation, Robust Control and Robotics, Springer-Verlag, London, 2001.
Karmarkar, M.: A New Polynomial-Time Algorithm for Linear Programming, Combinatorica 4 (1984), pp. 373–395.
Löfberg, J.: YALMIP, Yet Another LMI Parser, University of Linköping, Sweden, 2001, http://www.control.isy.liu.se/~johanl.
Nesterov, Y. and Nemirovskii, A.: Interior-Point Polynomial Methods in Convex Programming. SIAM, Philadelphia, 1994.
Neumaier, A.: Interval Methods for Systems of Equations, Cambridge University Press, Cambridge, 1990.
Rohn, J.: An Algorithm for Checking Stability of Symmetric Interval Matrices, IEEE Trans. Autom. Control 41 (1) (1996), pp. 133–136.
Sam-Haroud, D.: Constraint Consistency Techniques for Continuous Domains, PhD dissertation 1423, Swiss Federal Institute of Technology in Lausanne, 1995.
Scherer, C. and Weiland, S.: Course on LMIs in Control, Lecture Notes at Delft University of Technology and Eindhoven University of Technology, 2002. http://www.cs.ele.tue.nl/SWeiland/lmi.htm.
Sturm, J. F.: Using SeDuMi 1.02, a Matlab Toolbox for Optimization over Symmetric Cones, Optimization Methods and Software 11–12 (1999), pp. 625–653, http://fewcal.kub.nl/sturm.
van Hentenryck, P., Michel, L., and Deville, Y.: Numerica—A Modelling Language for Global Optimization, MIT Press, Cambridge, Massachusetts, 1997.
Vandenberghe, L. and Boyd, S.: Semidefinite Programming, SIAM Review 38 (1) (1996), pp. 49–95.
VanEmden, M.: Algorithmic Power from Declarative Use of Redundant Constraints, Constraints 4 (4) (1999), pp. 363–381.
Author information
Authors and Affiliations
Corresponding author
Additional information
D. Henrion acknowledges support of grant No. 102/02/0709 of the Grant Agency of the Czech Republic, and project No. ME 698/2003 of the Ministry of Education of the Czech Republic.
Rights and permissions
About this article
Cite this article
Jaulin, L., Henrion, D. Contracting Optimally an Interval Matrix without Loosing Any Positive Semi-Definite Matrix Is a Tractable Problem. Reliable Comput 11, 1–17 (2005). https://doi.org/10.1007/s11155-005-5939-3
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s11155-005-5939-3