# Ellipsoid Algorithms

Part of the Algorithms and Combinatorics book series (AC, volume 12)

## Abstract

Gaius Julius Caesar (101–44 B.C.) made military use of it when his legions conquered Gaulle, Machiavelli coined it as a political doctrine for Florentine princes to improve his princes’ control over their subjects, and mathematicians employ it e.g. when they use binary search to locate the optimum of a p-unimodal function over a compact convex subset of ℝ1. Divide and conquer, or more precisely divide and reign, is what Niccolo’s doctrine translates to. The compact convex subset of ℝ1 is, of course, some finite interval of the real line and the basic idea of binary search consists in successively halving this interval — like we did in Chapter 7.5.3. Utilizing the p-unimodality (see below) of the function to be optimized, we then discard one half of the original interval forever and continue the search in the other half of the original interval — which is again a compact convex subset of ℝ1. Consequently, the basic idea can be reapplied and we can iterate. Since the length of the left-over interval is half of the length of the previous interval, the 1-dimensional “volume” of the remaining compact convex subset of ℝ1 to be searched shrinks at a geometric rate and we obtain fast convergence of the iterative scheme.

### Keywords

Hull Hunt Lution Marquis

## Preview

### References

1. Bland, R.G., D. Goldfarb and M.J. Todd [ 1981 ] “The ellipsoid method: a survey”, Operations Research 29 1039–1091.
2. Gâcs, P. and L. Lovasz [ 1981 ] “Khachian’s algorithm for linear programming”, Mathematical Programming Study 14 61–68.
3. Grötschel, M., L. Lovâsz and A. Schrijver [ 1981 ] “The ellipsoid method and its consequences in combinatorial optimization”, Combinatorica 1 169–197.
4. Grötschel, M., L. Lovâsz and A. Schrijver [ 1988 ] Geometric Algorithms and Combinatorial Optimization, Springer-Verlag, Berlin.
5. Karp, R.M. and C.H. Papadimitriou [ 1982 ] “On linear characterization of combinatorial optimization problems” SIAM Journal on Computing 11 620–632.
6. Khachian, L.G. [ 1979 ] “A polynomial algorithm in linear programming”, Soviet Mathematics Doklady 20 191–194.Google Scholar
7. Khachian, L.G. [ 1980 ] “Polynomial algorithms in linear programming”, USSR Comp. Math. and Math. Phys. 20 53–72.
8. Khachian, L.G. [ 1982 ] “On exact solutions of systems of linear inequalities and LP problems”, USSR Comp. Math. and Math. Phys. 22 239–242.
9. Khachian, L.G. [ 1982 ] “Convexity and computational complexity in polynomial programming”, Engineering Cybernetics 22 46–56.Google Scholar
10. Kiefer, J. [ 1952 ] “Sequential minimax search for a maximum”, Proc. Amer. Math. Soc. 4 502–506.
11. Korte, B. and R. Schrader [ 1980 ] “A note on convergence proofs for Shor-Khachian methods”, in Auslender, Oettli, Stoer (eds) Optimization and Optimal Control, Lecture Notes in Control and Information Sciences Vol. 30, Springer, Berlin, New York, 51–57.Google Scholar
12. Levin, A.Y. [ 1965 ] “On an algorithm for convex function minimization”, Soviet Mathematics Doklady 6 286–290.Google Scholar
13. Newman, D.J. [ 1965 ] “Location of the maximum on unimodal surfaces”, Journal of the Association for Computing Machinery 12 395–398.
14. Padberg, M. and M.R.Rao [ 1980 ] “The Russian method: Part I, Part I I, Part III”, Graduate School of Business Administration, New York University, New York.Google Scholar
15. Padberg, M. and G. Rinaldi [ 1991 ] “A branch-and-cut algorithm for the resolution of large-scale symmetric traveling salesman problems”, SIAM Review 33 60–100.
16. Schrader, R. [ 1982 ] “Ellipsoid methods”, in D. Korte (ed) Modern Applied Mathematics–Optimization and Operations Research, North-Holland, Amsterdam, 265–311.Google Scholar
17. Schrijver, A. [ 1986 ] Theory of Linear and Integer Programming, Wiley, Chichester.
18. Shor, N.Z. [ 1970 ] “Utilization of the operation of space dilatation in the minimization of convex functions”, Cybernetics 6 7–15.
19. Shor, N.Z. [ 1970 ] “Convergence rate of the gradient method with dilatation of the space”, Cybernetics 6 102–108.
20. Shor, N.Z. [1977] “Cut-off method with space extension in convex programming problems, Cybernetics 13 94–96.Google Scholar
21. Wolfe, P. [ 1980 ] “A bibliography for the ellipsoid algorithm”, IBM Research Center, Yorktown Heights, NY.Google Scholar
22. Yudin, D.B. and Nemirovskii, A.S. [ 1976 ] “Informational complexity and efficient methods for the solution of convex extremal problems” Matekon 13 3–25.Google Scholar
23. Golub, G.H. and C.F.Van Loan [ 1983 ] Matrix Computations, The Johns Hopkins University Press, Baltimore.
24. Ostrowski, A.M. [ 1973 ] Solution of equations in Euclidean and Banach spaces, 3rd Ed., Academic Press, New York.
25. Grötschel, M., L. Lovâsz and A. Schrijver [ 1988 ] Geometric Algorithms and Combinatorial Optimization, Springer-Verlag, Berlin.
26. John, F. [ 1948 ] “Extremum problems with inequalities as subsidiary conditions” in Studies and Essays, Courant anniversary volume, New York, Interscience.Google Scholar
27. Khachian, L.G. [ 1979 ] “A polynomial algorithm in linear programming”, Soviet Mathematics Doklady 20 191–194.Google Scholar
28. Padberg, M. and M.R.Rao [ 1979 ] “The Russian method for linear inequalities and linear optimization”, November 1979, Revised June 1980, Graduate School of Business Administration, New York University, New York.Google Scholar
29. Padberg, M. and M.R.Rao [ 1979 ] “The Russian method for linear inequalities II: Approximate arithmetic”, January 1980, Graduate School of Business Administration, New York University, New York.Google Scholar
30. Shor, N.Z. [ 1970 ] “Utilization of the operation of space dilatation in the minimization of convex functions”, Cybernetics 6 7–15.
31. Shor, N.Z. [ 1970 ] “Convergence rate of the gradient method with dilatation of the space”, Cybernetics 6 102–108.
32. Dantzig, G.B., D.R. Fulkerson and S.M. Johnson [ 1954 ] “Solution of a large-scale travelling salesman problem”, Operations Research 2 393–410.
33. Padberg, M. and M.R. Rao [ 1982 ] “Odd Minimum Cut-Sets and b-Matchings”, Mathematics of Operations Research 7 67–80.
34. Padberg, M. and G. Rinaldi [ 1991 ] “A branch-and-cut algorithm for the resolution of large-scale symmetric traveling salesman problems”, SIAM Review 33 60–100.
35. Cassels, J.W.S. [ 1965 ] An Introduction to Diophantine Approximation, Cambridge University Press, Cambridge.Google Scholar
36. Courant, R. and H. Robbins [ 1978 ] What is Mathematics, Oxford University Press, New York.
37. Grötschel, M., L. Lovâsz and A. Schrijver [ 1988 ] Geometric Algorithms and Combinatorial Optimization, Springer-Verlag, Berlin.
38. Grünbaum, B. [ 1967 ] Convex Polytopes, Wiley, London.
39. Khinchin, A.Y. [ 1935 ] Continued Fractions (in Russian). English translation [1964], The University of Chicago Press, Chicago, Illinois.Google Scholar
40. Stoer, J. and C. Witzgall [ 1970 ] Convexity and Optimization in Finite Dimensions I, Springer, Berlin.