Abstract
Constraints are often represented as ellipsoids. One of the main advantages of such constrains is that, in contrast to boxes, over which optimization of even quadratic functions is NP-hard, optimization of a quadratic function over an ellipsoid is feasible. Sometimes, the area described by constrains is too large, so it is reasonable to bisect this area (one or several times) and solve the optimization problem for all the sub-areas. Bisecting a box, we still get a box, but bisecting an ellipsoid, we do not get an ellipsoid. Usually, this problem is solved by enclosing the half-ellipsoid in a larger ellipsoid, but this slows down the domain reduction process. Instead, we propose to optimize the objective functions over the resulting half-, quarter, etc., ellipsoids.
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References
Belforte, G., Bona, B.: An improved parameter identification algorithm for signal with unknown-but-bounded errors. In: Proceedings of the 7th IFAC Symposium on Identification and Parameter Estimation, NewYork, U.K. (1985)
Chernousko, F.L.: Estimation of the Phase Space of Dynamic Systems. Nauka Publ., Moscow (1988) (in Russian)
Chernousko, F.L.: State Estimation for Dynamic Systems. CRC Press, Boca Raton (1994)
Cormen, C.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms. MIT Press, Boston (2009)
Ferson, S., Ginzburg, L., Kreinovich, V., Longpré, L., Aviles, M.: Exact bounds on finite populations of interval data. Reliable Computing 11(3), 207–233 (2005)
Filippov, A.F.: Ellipsoidal estimates for a solution of a system of differential equations. Interval Computations 2(2(4)), 6–17 (1992)
Fogel, E., Huang, Y.F.: On the value of information in system identification. Bounded noise case. Automatica 18(2), 229–238 (1982)
Interval computations website, http://www.cs.utep.edu/interval-comp
Jaulin, L., Kieffer, M., Didrit, O., Walter, E.: Applied Interval Analysis, with Examples in Parameter and State Estimation, Robust Control and Robotics. Springer, London (2001)
Karmarkar, N.: A new polynomial-time algorithm for linear programming. Combinatorica 4, 373–396 (1984)
Kreinovich, V., Lakeyev, A., Rohn, J., Kahl, P.: Computational Complexity and Feasibility of Data Processing and Interval Computations. Kluwer, Dordrecht (1998)
Moore, R.E., Kearfott, R.B., Cloud, M.J.: Introduction to Interval Analysis. SIAM Press, Philadelphia (2009)
Schweppe, F.C.: Recursive state estimation: unknown but bounded errors and system inputs. IEEE Transactions on Automatic Control 13, 22 (1968)
Schweppe, F.C.: Uncertain Dynamic Systems. Prentice Hall, Englewood Cliffs (1973)
Vavasis, S.A.: Nonlinear Optimization: Complexity Issues. Oxford University Press, New York (1991)
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Portillo, P., Ceberio, M., Kreinovich, V. (2014). Towards an Efficient Bisection of Ellipsoids. In: Ceberio, M., Kreinovich, V. (eds) Constraint Programming and Decision Making. Studies in Computational Intelligence, vol 539. Springer, Cham. https://doi.org/10.1007/978-3-319-04280-0_16
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DOI: https://doi.org/10.1007/978-3-319-04280-0_16
Publisher Name: Springer, Cham
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