A deep cut ellipsoid algorithm for convex programming: Theory and applications

This paper proposes a deep cut version of the ellipsoid algorithm for solving a general class of continuous convex programming problems. In each step the algorithm does not require more computational effort to construct these deep cuts than its corresponding central cut version. Rules that prevent some of the numerical instabilities and theoretical drawbacks usually associated with the algorithm are also provided. Moreover, for a large class of convex programs a simple proof of its rate of convergence is given and the relation with previously known results is discussed. Finally some computational results of the deep and central cut version of the algorithm applied to a min—max stochastic queue location problem are reported.

[1]  Daniel P. Heyman,et al.  Stochastic models in operations research , 1982 .

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

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

[4]  Michael J. Todd,et al.  The Ellipsoid Method: A Survey , 1980 .

[5]  T. Ibaraki,et al.  The Computational Complexity of the m -Center Problems on the Plane , 1981 .

[6]  Leonard Kleinrock,et al.  Theory, Volume 1, Queueing Systems , 1975 .

[7]  Shuzhong Zhang,et al.  A deep cut ellipsoid algorithm and quasiconvex programming , 1994 .

[8]  D. Varberg.  Convex Functions , 1973 .

[9]  Michael Ian Shamos,et al.  Closest-point problems , 1975, 16th Annual Symposium on Foundations of Computer Science (sfcs 1975).

[10]  K. Nair,et al.  Optimal location of a single service center of certain types , 1971 .

[11]  Claude Dennis Pegden,et al.  Nonlinear Programming Theory and Algorithms , 1979 .

[12]  Martin Grötschel,et al.  The Ellipsoid Method , 1993 .

[13]  Nimrod Megiddo,et al.  Linear-time algorithms for linear programming in R3 and related problems , 1982, 23rd Annual Symposium on Foundations of Computer Science (sfcs 1982).

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

[15]  N. Megiddo.  Linear-time algorithms for linear programming in R3 and related problems , 1982, FOCS 1982.

[16]  Mokhtar S. Bazaraa,et al.  Nonlinear Programming: Theory and Algorithms , 1993 .

[17]  C. O. Oakley,et al.  The Enjoyment of Mathematics. , 1957 .

[18]  Nimrod Megiddo,et al.  Linear-Time Algorithms for Linear Programming in R^3 and Related Problems , 1982, FOCS.

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

[20]  Claude George.  Exercises in integration , 1984 .

[21]  V. M. Volosov.  Some types of calculation connected with averaging in the theory of non-linear vibrations , 1963 .

[22]  J. Ecker,et al.  A Computational Comparison of the Ellipsoid Algorithm with Several Nonlinear Programming Algorithms , 1985 .

[23]  Michael Kupferschmid,et al.  Using deep cuts in an ellipsoid algorithm for nonlinear programming , 1985 .

[24]  A.Ya. Dubovitskii,et al.  Extremum problems in the presence of restrictions , 1965 .

[25]  Michael Kupferschmid,et al.  An ellipsoid algorithm for nonlinear programming , 1983, Math. Program..

[26]  Laurence A. Wolsey,et al.  Integer and Combinatorial Optimization , 1988 .

[27]  Y. Ye.  Karmarkar's algorithm and the ellipsoid method , 1987 .

[28]  Shuzhong Zhang,et al.  An Efficient Algorithm to Check Whether O Belongs to the Convex Hull of Finite Number of Lp-Circles , 1991 .

[29]  Onno J. Boxma,et al.  Workloads and waiting times in single-server systems with multiple customer classes , 1989, Queueing Syst. Theory Appl..

[30]  F. R. Gantmakher.  The Theory of Matrices , 1984 .

[31]  Jean-Louis Goffin,et al.  Variable metric relaxation methods, part II: The ellipsoid method , 1984, Math. Program..

[32]  D. Hearn,et al.  Geometrical Solutions for Some Minimax Location Problems , 1972 .

[33]  L. Lovász,et al.  Geometric Algorithms and Combinatorial Optimization , 1981 .

[34]  Jean-Louis Goffin,et al.  Convergence Rates of the Ellipsoid Method on General Convex Functions , 1983, Math. Oper. Res..

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

[36]  L. Khachiyan.  Polynomial algorithms in linear programming , 1980 .

[37]  Hans-Jakob Lüthi,et al.  On the Solution of Variational Inequalities by the Ellipsoid Method , 1985, Math. Oper. Res..

[38]  Oded Berman,et al.  Optimal Server Location on a Network Operating as an M/G/1 Queue , 1985, Oper. Res..

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

[40]  Michael J. Todd,et al.  Feature Article - The Ellipsoid Method: A Survey , 1981, Oper. Res..

[41]  James E. Ward,et al.  Using Block Norms for Location Modeling , 1985, Oper. Res..

[42]  G. Hardy,et al.  What is Mathematics? , 1942, Nature.

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

[44]  Richard D. Smallwood,et al.  Minimax Detection Station Placement , 1965 .