Minimal ellipsoid circumscribing a polytope defined by a system of linear inequalities

Journal of Global Optimization

Volumn 34, Issue 1, 2006, Pages 1-14

  Gotoh, Jun Ya ^a   Konno, Hiroshi ^b  

^a UNIVERSITY OF TSUKUBA   (Japan)

^b CHUO UNIVERSITY   (Japan)

Author keywords

Computational geometry; Global optimization; Minimal ellipsoid

Indexed keywords

ALGORITHMS; GLOBAL OPTIMIZATION; ITERATIVE METHODS; LINEAR SYSTEMS; PROBLEM SOLVING; SET THEORY;

LINEAR INEQUALITIES; MINIMAL ELLIPSOIDS; POLYTOPES;

COMPUTATIONAL GEOMETRY;

EID: 29144457300     PISSN: 09255001     EISSN: 15732916     Source Type: Journal    
DOI: 10.1007/s10898-005-3883-8     Document Type: Article

References (22)

1. Barnes, E.R. (1982), An algorithm for separating patterns by ellipsoids, IBM Journal of Research Development 26(6), 759-764.
2. Ben-Tal, A. and Nemirovski, A. (2001), Lectures on Modern Convex Optimization -Analysis, Algorithms, and Engineering Applications-, SIAM.
3. Dyer, M.E. (1986), On a multidimensional search technique and its applications to the euclidean one-center problem, SIAM Journal of Computing, 15 725-738.
4. Elzinga, J. and Hearn, D.W (1972), The minimal covering sphere problem, Management Science, 19, 96-104.
5. Fukuda, K. (1995), cdd/cdd+ Reference Manual, Institute for Operations Research, Swiss Federal Institute of Technology, Zurich, Switzerland, program available from http://www.ifor.math.ethz.ch/fukuda/fukuda.html
6. Falk, J. and Soland, R.M. (1969), An algorithm for separable nonconvex programming problems, Management Science 15, 550-569.
7. Gärtner, B. and Schönherr, S. (1997), Smallest enclosing ellipses -fast and exact-, Proceedings 13. Annual ACM Symposium on Computational Geometry 430-432.
8. John, F. (1948), Extremum problems with inequalities as subsidiary conditions, in: Studiesf and Essays Presented to R. Courant on His Birthday, Interscience Publishers, NY, pp. 187-204.
9. Khachiyan, L. and Todd, M. (1993), On the complexity of approximating the maximal inscribed ellipsoid for a polytope, Mathematical Programming 61, 137-159.
10. Konno, H., Thach, P.T. and Tuy, H. (1997), Optimization on low rank nonconvex structures, Kluwer Academic Publishers, Dordrecht.
11. Konno, H. Yajima, Y. and Ban, A. (1994), Calculating a minimal sphere containing a polytope defined by a system of linear inequalities, Computational Optimization and Applications, 3, 181-191.
12. Konno, H. and Wijayanayake, A. (1999), Mean-absolute deviation portfolio optimization model under transaction costs, Journal of the Operations Research Society of Japan 42, 422-435.
13. Konno, H. and Wijayanayake, A. (2001), Portfolio optimization problem under concave transaction costs and minimal transaction unit constraints, Mathematical Programming B89, 233-250.
14. Phong, T.Q., An, L.T.H. and Tao, P.D. (1995), On globally solving linearly constrained indefinite quadratic minimization problems by decomposition branch and bound method, Operations Research Letters 17, 215-220.
15. Rockafellar, R.T. and Wets, R.J-B. (1998), Variational Analysis, Springer-Verlag Berlin.
16. Sekitani, K. and Yamamoto, Y. (1993), A recursive algorithm for finding the minimal covering sphere of a polytope and the minimal covering spheres of several polytopes, Japan Journal of Industrial and Applied Mathematics 10, 255-273.
17. Skyum, S. (1991), A simple algorithm for computing the smallest enclosing circle, Information Processing Letters 37, 121-125.
18. Sun, P. and Freund, R.M. (2004), Computation of minimum volume covering Ellipsoids, Operations Research 52, 690-706.
19. Tuy, H. (1998), Convex Analysis and Global Optimization, Kluwer Academic Publishers, Dordrecht.
20. Welzl, E. (1991), Smallest enclosing disks (Balls and Ellipsoids), in: Maurer, H. (ed.), New Results and New Trends in Computer Science, of Lecture Notes in Computer Science, Springer-Verlag, Berlin, vol. 555. pp. 359-370.
21. Zhang, Y. (1998), An interior-point algorithm for the maximum-volume Ellipsoid problem, Department of Computational and Applied Mathematics, Rice University, Technical Report TR98-15.
22. Zhang, Y. and Gao, L. (2003), On numerical solution of the maximum volume ellipsoid problem, SIAM Journal of Optimization 14(1), 53-76.