Linear Optimization and Extensions pp 309-386 | Cite as

# Ellipsoid Algorithms

## 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 ℝ^{1}.
*Divide and conquer*, or more precisely divide and reign, is what Niccolo’s doctrine translates to. The compact convex subset of ℝ^{1} 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 ℝ^{1}. 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 ℝ^{1} to be searched shrinks at a geometric rate and we obtain fast convergence of the iterative scheme.

## Keywords

Extreme Point Rational Vector Compact Convex Subset Linear Optimization Problem Asymptotic Cone## Preview

Unable to display preview. Download preview PDF.

## References

- Bland, R.G., D. Goldfarb and M.J. Todd [ 1981 ] “The ellipsoid method: a survey”,
*Operations Research*29 1039–1091.MathSciNetMATHCrossRefGoogle Scholar - Gâcs, P. and L. Lovasz [ 1981 ] “Khachian’s algorithm for linear programming”,
*Mathematical Programming Study*14 61–68.MATHCrossRefGoogle Scholar - Grötschel, M., L. Lovâsz and A. Schrijver [ 1981 ] “The ellipsoid method and its consequences in combinatorial optimization”,
*Combinatorica*1 169–197.MathSciNetMATHCrossRefGoogle Scholar - Grötschel, M., L. Lovâsz and A. Schrijver [ 1988 ]
*Geometric Algorithms and Combinatorial Optimization*, Springer-Verlag, Berlin.MATHCrossRefGoogle Scholar - Karp, R.M. and C.H. Papadimitriou [ 1982 ] “On linear characterization of combinatorial optimization problems”
*SIAM Journal on Computing*11 620–632.MathSciNetMATHCrossRefGoogle Scholar - Khachian, L.G. [ 1979 ] “A polynomial algorithm in linear programming”,
*Soviet Mathematics Doklady*20 191–194.Google Scholar - Khachian, L.G. [ 1980 ] “Polynomial algorithms in linear programming”,
*USSR Comp. Math. and Math. Phys.*20 53–72.CrossRefGoogle Scholar - Khachian, L.G. [ 1982 ] “On exact solutions of systems of linear inequalities and LP problems”,
*USSR Comp. Math. and Math. Phys.*22 239–242.CrossRefGoogle Scholar - Khachian, L.G. [ 1982 ] “Convexity and computational complexity in polynomial programming”,
*Engineering Cybernetics*22 46–56.Google Scholar - Kiefer, J. [ 1952 ] “Sequential minimax search for a maximum”,
*Proc. Amer. Math. Soc.*4 502–506.MathSciNetCrossRefGoogle Scholar - 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 - Levin, A.Y. [ 1965 ] “On an algorithm for convex function minimization”,
*Soviet Mathematics Doklady*6 286–290.Google Scholar - Newman, D.J. [ 1965 ] “Location of the maximum on unimodal surfaces”,
*Journal of the Association for Computing Machinery*12 395–398.MathSciNetMATHCrossRefGoogle Scholar - 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
- 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.MathSciNetMATHCrossRefGoogle Scholar - Schrader, R. [ 1982 ] “Ellipsoid methods”, in D. Korte (ed)
*Modern Applied Mathematics–Optimization and Operations Research*, North-Holland, Amsterdam, 265–311.Google Scholar - Schrijver, A. [ 1986 ]
*Theory of*Linear*and Integer Programming*, Wiley, Chichester.MATHGoogle Scholar - Shor, N.Z. [ 1970 ] “Utilization of the operation of space dilatation in the minimization of convex functions”,
*Cybernetics*6 7–15.CrossRefGoogle Scholar - Shor, N.Z. [ 1970 ] “Convergence rate of the gradient method with dilatation of the space”,
*Cybernetics*6 102–108.CrossRefGoogle Scholar - Shor, N.Z. [1977] “Cut-off method with space extension in convex programming problems,
*Cybernetics*13 94–96.Google Scholar - Wolfe, P. [ 1980 ] “A bibliography for the ellipsoid algorithm”, IBM Research Center, Yorktown Heights, NY.Google Scholar
- 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 - Golub, G.H. and C.F.Van Loan [ 1983 ]
*Matrix Computations*, The Johns Hopkins University Press, Baltimore.MATHGoogle Scholar - Ostrowski, A.M. [ 1973 ]
*Solution of equations in Euclidean and Banach spaces*, 3rd Ed., Academic Press, New York.MATHGoogle Scholar - Grötschel, M., L. Lovâsz and A. Schrijver [ 1988 ]
*Geometric Algorithms and Combinatorial Optimization*, Springer-Verlag, Berlin.MATHCrossRefGoogle Scholar - John, F. [ 1948 ] “Extremum problems with inequalities as subsidiary conditions” in
*Studies and Essays*, Courant anniversary volume, New York, Interscience.Google Scholar - Khachian, L.G. [ 1979 ] “A polynomial algorithm in linear programming”,
*Soviet Mathematics Doklady*20 191–194.Google Scholar - 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
- 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
- Shor, N.Z. [ 1970 ] “Utilization of the operation of space dilatation in the minimization of convex functions”,
*Cybernetics*6 7–15.CrossRefGoogle Scholar - Shor, N.Z. [ 1970 ] “Convergence rate of the gradient method with dilatation of the space”,
*Cybernetics*6 102–108.CrossRefGoogle Scholar - Dantzig, G.B., D.R. Fulkerson and S.M. Johnson [ 1954 ] “Solution of a large-scale travelling salesman problem”,
*Operations Research*2 393–410.MathSciNetCrossRefGoogle Scholar - Padberg, M. and M.R. Rao [ 1982 ] “Odd Minimum Cut-Sets and b-Matchings”,
*Mathematics of Operations Research*7 67–80.MathSciNetMATHCrossRefGoogle Scholar - 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.MathSciNetMATHCrossRefGoogle Scholar - Cassels, J.W.S. [ 1965 ]
*An Introduction to Diophantine Approximation*, Cambridge University Press, Cambridge.Google Scholar - Courant, R. and H. Robbins [ 1978 ]
*What is Mathematics*, Oxford University Press, New York.MATHGoogle Scholar - Grötschel, M., L. Lovâsz and A. Schrijver [ 1988 ]
*Geometric Algorithms and Combinatorial Optimization*, Springer-Verlag, Berlin.MATHCrossRefGoogle Scholar - Grünbaum, B. [ 1967 ]
*Convex Polytopes*, Wiley, London.MATHGoogle Scholar - Khinchin, A.Y. [ 1935 ]
*Continued Fractions*(in Russian). English translation [1964], The University of Chicago Press, Chicago, Illinois.Google Scholar - Stoer, J. and C. Witzgall [ 1970 ]
*Convexity and Optimization in Finite Dimensions I*, Springer, Berlin.CrossRefGoogle Scholar