Feature Article - The Ellipsoid Method: A Survey

In February 1979 a note by L. G. Khachiyan indicated how an ellipsoid method for linear programming can be implemented in polynomial time. This result has caused great excitement and stimulated a flood of technical papers. Ordinarily there would be no need for a survey of work so recent, but the current circumstances are obviously exceptional. Word of Khachiyan's result has spread extraordinarily fast, much faster than comprehension of its significance. A variety of issues have, in general, not been well understood, including the exact character of the ellipsoid method and of Khachiyans result on polynomiality, its practical significance in linear programming, its implementation, its potential applicability to problems outside of the domain of linear programming, and its relationship to earlier work. Our aim is to help clarify these important issues in the context of a survey of the ellipsoid method, its historical antecedents, recent developments, and current research.

[1]  L. G. H. Cijan.  A polynomial algorithm in linear programming , 1979 .

[2]  P. Gill,et al.  Methods for computing and modifying the $LDV$ factors of a matrix , 1975 .

[3]  William Y. Sit,et al.  Worst case behavior of the steepest edge simplex method , 1979, Discret. Appl. Math..

[4]  Norman Zadeh,et al.  A bad network problem for the simplex method and other minimum cost flow algorithms , 1973, Math. Program..

[5]  Yehoshua Bar-Hillel,et al.  The Intrinsic Computational Difficulty of Functions , 1969 .

[6]  J. Edmonds.  Paths, Trees, and Flowers , 1965, Canadian Journal of Mathematics.

[7]  D. J. Newman,et al.  Location of the Maximum on Unimodal Surfaces , 1965, JACM.

[8]  J. Scott Provan,et al.  An Application of the KhachianShor Algorithm to a Class of Linear Complementary Problems , 1980 .

[9]  P. Gács,et al.  Khachiyan’s algorithm for linear programming , 1981 .

[10]  Jan Telgen,et al.  On relaxation methods for systems of linear inequalities , 1982 .

[11]  T. C. Hu,et al.  Synthesis of a Communication Network , 1964 .

[12]  T. C. Hu,et al.  An Application of Generalized Linear Programming to Network Flows , 1962 .

[13]  T. C. Hu,et al.  Multi-Terminal Network Flows , 1961 .

[14]  E. T..  An Introduction to the Theory of Numbers , 1946, Nature.

[15]  N. Z. Shor.  Convergence rate of the gradient descent method with dilatation of the space , 1970 .

[16]  Philip Wolfe,et al.  Note on a method of conjugate subgradients for minimizing nondifferentiable functions , 1974, Math. Program..

[17]  Martin Grötschel,et al.  The ellipsoid method and its consequences in combinatorial optimization , 1981, Comb..

[18]  I. J. Schoenberg,et al.  The Relaxation Method for Linear Inequalities , 1954, Canadian Journal of Mathematics.

[19]  George B. Dantzig,et al.  Linear programming and extensions , 1965 .

[20]  Richard M. Karp,et al.  On linear characterizations of combinatorial optimization problems , 1980, 21st Annual Symposium on Foundations of Computer Science (sfcs 1980).

[21]  Boris Polyak.  Minimization of unsmooth functionals , 1969 .

[22]  Eugene L. Lawler.  The Great Mathematical Sputnik of 1979 , 1980 .

[23]  Stephen A. Cook,et al.  The complexity of theorem-proving procedures , 1971, STOC.

[24]  V. A. Skokov.  Note on minimization methods employing space stretching , 1974 .

[25]  V. Chvátal,et al.  Notes on Bland’s pivoting rule , 1978 .

[26]  C. Lemaréchal.  An extension of davidon methods to non differentiable problems , 1975 .

[27]  D. R. Fulkerson,et al.  Flows in Networks. , 1964 .

[28]  S. N. Maheshwari,et al.  An O(|V|³) Algorithm for Finding Maximum Flows in Networks , 1978, Inf. Process. Lett..

[29]  M. Todd,et al.  Modifications and Implementation of the Shor-Khachian Algorithm for Linear Programming , 1980 .

[30]  H. König,et al.  On Khachian's algorithm and minimal ellipsoids , 1980 .

[31]  Richard M. Karp,et al.  On the Computational Complexity of Combinatorial Problems , 1975, Networks.

[32]  A. Hoffman.  On approximate solutions of systems of linear inequalities , 1952 .

[33]  L. Lovász.  A new linear programming algorithm — better or worse than the Simplex Method? , 1980 .

[34]  V. Klee,et al.  HOW GOOD IS THE SIMPLEX ALGORITHM , 1970 .

[35]  Alfred V. Aho,et al.  The Design and Analysis of Computer Algorithms , 1974 .

[36]  W. H. Cunningham,et al.  Theoretical Properties of the Network Simplex Method , 1979, Math. Oper. Res..

[37]  M. V. Novikov,et al.  Academy of Sciences of the Ukrainian SSR , 1970 .

[38]  N. Z. Shor.  The rate of convergence of the generalized gradient descent method , 1968 .

[39]  L. Khachiyan,et al.  The polynomial solvability of convex quadratic programming , 1980 .

[40]  Nesa L'abbe Wu,et al.  Linear programming and extensions , 1981 .

[41]  V. Klee,et al.  Helly's theorem and its relatives , 1963 .

[42]  N. Z. Shor.  Utilization of the operation of space dilatation in the minimization of convex functions , 1972 .

[43]  Robert G. Jeroslow,et al.  The simplex algorithm with the pivot rule of maximizing criterion improvement , 1973, Discret. Math..

[44]  S. Agmon.  The Relaxation Method for Linear Inequalities , 1954, Canadian Journal of Mathematics.

[45]  N. Z. Shor.  Cut-off method with space extension in convex programming problems , 1977, Cybernetics.

[46]  N. Z. Shor,et al.  A minimization method using the operation of extension of the space in the direction of the difference of two successive gradients , 1971 .

[47]  P. Wolfe.  Note on a method of conjugate subgradients for minimizing nondifferentiable functions , 1974 .

[48]  R. Bartels.  A stabilization of the simplex method , 1971 .

[49]  N. Shor.  New development trends in nondifferentiable optimization , 1977 .

[50]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[51]  N. Z. Shor,et al.  Family of algorithms for solving convex programming problems , 1979 .

[52]  Michael J. Todd.  On Minimum Volume Ellipsoids Containing Part of a Given Ellipsoid , 1982, Math. Oper. Res..

[53]  Richard M. Karp,et al.  The traveling-salesman problem and minimum spanning trees: Part II , 1971, Math. Program..

[54]  Yu. M. Ermol’ev.  Methods of solution of nonlinear extremal problems , 1966 .

[55]  Richard M. Karp,et al.  The Traveling-Salesman Problem and Minimum Spanning Trees , 1970, Oper. Res..