On Vaidya's volumetric cutting plane method for convex programming

Mathematics of Operations Research

Volumn 22, Issue 1, 1997, Pages 63-89

  Anstreicher, Kurt M ^a  

^a UNIVERSITY OF IOWA   (United States)

Author keywords

Convex programming; Cutting plane method; Ellipsoid algorithm; Interior point algorithm; Linear programming; Volumetric barrier

Indexed keywords

ALGORITHMS; COMPUTATIONAL COMPLEXITY; CONSTRAINT THEORY; PROBLEM SOLVING; SET THEORY;

CONVEX PROGRAMMING; CUTTING PLANE METHOD; ELLIPSOID METHOD; INTERIOR POINT METHODS; LINEAR INEQUALITY CONSTRAINTS; VOLUMETRIC BARRIER;

LINEAR PROGRAMMING;

EID: 0031071780     PISSN: 0364765X     EISSN: None     Source Type: Journal    
DOI: 10.1287/moor.22.1.63     Document Type: Article

References (30)

1. Altman, A., K. C. Kiwiel (1996). A note on some analytic cutting plane methods for convex feasibility and minimization problems. Computational Optim. Appl. 5 175-180.
2. Anstreicher, K. M. (1996). Large step volumetric potential reduction algorithms for linear programming. Ann. Oper. Res. 62 521-538.
3. _ (1997). Volumetric path following algorithms for linear programming. Math. Programming 76 245-263.
4. Atkinson, D. S., P. M. Vaidya (1995). A cutting plane algorithm for convex programming that uses analytic centers. Math. Programming 69 1-43.
5. Bahn, O., O. Du Merle, J.-L. Goffin, J.-P. Vial (1995). A cutting plane method from analytic centers for stochastic programming. Math. Programming 69 45-73.
6. _ J.-L. Goffin, J.-P. Vial, O. Du Merle (1994). Experimental behavior of an interior point cutting plane algorithm for convex programming: An application to geometric programming. Discrete Appl. Math. 49 3-23.
7. Bixby, R. E., J. W. Gregory, I. J. Lustig, R. E. Marsten, D. F. Shanno (1992). Very large-scale linear programming: A case study in combining interior point and simplex methods. Oper. Res. 40 885-897.
8. Bland, R. G., D. Goldfarb, M. J. Todd (1981). The ellipsoid method: A survey. Oper. Res. 29 1039-1091.
9. Coppersmith, D., S. Winograd (1986). Matrix multiplication via arithmetic progression, Proceedings of the 19th Annual ACM Symposium on Theory of Computing, 1-6.
10. den Hertog, D., C. Roos, T. Terlaky (1992), A build-up variant of the path-following method for LP. Oper. Res. Lett. 12 181-186.
11. Goffin, J.-L. (1994). Using the primal dual infeasible Newton method in the analytic center method for problems defined by deep cutting planes. Faculty of Management, McGill University, Montreal, Canada.
12. _, A. Haurie, J.-P. Vial (1992). Decomposition and nondifferentiable optimization with the projective algorithm. Management Sci. 38 284-302.
13. _, Z.-Q. Luo, Y. Ye (1993). Complexity analysis of an interior point cutting plane method for convex feasibility problems. GERAD, Faculty of Management, McGill University, Montreal, Canada SIAM J. Optim. (to appear).
14. Grötschel, M., L. Lovasz, A. Schrijver (1988). Geometric Algorithms and Combinatorial Optimization. Springer-Verlag, Berlin.
15. Horn, R. A., C. R. Johnson (1985), Matrix Analysis. Cambridge University Press, Cambridge.
16. Kaliski, J. A., Y. Ye (1993). A decomposition variant of the potential reduction algorithm for linear programming, Management Sci. 39 757-776.
17. Luo, Z.-Q. (1994). Analysis of a cutting plane method that uses weighted analytic centers and multiple cuts. Department of Electrical and Computer Engineering, McMaster University, Hamilton, Ontario, Canada.
18. Mitchell, J. E., B. Borchers (1996). Solving real-world linear ordering problems using a primal-dual interior point cutting plane method. Ann. Oper. Res. 62 253-276.
19. _, S. Ramaswamy (1993). An extension of Atkinson and Vaidya's algorithm that uses the central trajectory. Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY
20. _, M. J. Todd (1992). Solving combinatorial optimization problems using Karmarkar's algorithm. Math. Programming 56 245-284.
21. Nemirovskii, A. S., D. B. Yudin (1983). Problem Complexity and Method Efficiency in Optimization. John Wiley, Chichester.
22. Nesterov, Y. (1995). Complexity estimates of some cutting plane methods based on analytical barrier. Math. Programming 69 149-176.
23. _, A. Nemirovskii (1994). Interior-Point Polynomial Algorithms in Convex Programming. SIAM, Philadelphia.
24. Ramaswamy, S., J. E. Mitchell (1995). A long step cutting plane algorithm that uses the volumetric barrier. Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY.
25. Sonnevend, G. (1988). New algorithms in convex programming based on a notion of "centre" (for systems of analytic inequalities) and on rational extrapolation, K. H. Hoffmann et al., eds. Trends in Mathematical Optimization: Proceedings of the 4th French-German Conference on Optimization, Internat. Ser. Numer. Math. 84 311-327.
26. Vaidya, P. M. (1996). A new algorithm for minimizing convex functions over convex sets. Math. Programming 73 291-341.
27. 1.5n)L) arithmetic operations. Math. Programming 47 175-201.
28. _, D. S. Atkinson (1993). A technique for bounding the number of iterations in path following algorithms, P. M. Pardalos, ed. Complexity in Numerical Optimization, World Scientific, Singapore, 462-489.
29. Ye, Y. (1992). A potential reduction algorithm allowing column generation. SIAM J. Optim. 2 7-20.
30. _ (1994). Complexity analysis of the analytic center cutting plane method that uses multiple cuts. Department of Management Sciences, University of Iowa, Iowa City, IA.