Polynomial interior point cutting plane methods

Optimization Methods and Software

Volumn 18, Issue 5, 2003, Pages 507-534

  Mitchell, John E ^a  

^a Rensselaer Polytechnic Institute   (United States)

Author keywords

Analytic center; Convex optimization; Cutting plane methods; Nonsmooth optimization; Volumetric center

Indexed keywords

ALGORITHMS; LINEAR PROGRAMMING; OPTIMIZATION; SET THEORY;

POLYNOMIAL CUTTING PLANE METHODS;

POLYNOMIALS;

EID: 0242551555     PISSN: 10556788     EISSN: None     Source Type: Journal    
DOI: 10.1080/10556780310001607956     Document Type: Article

References (37)

1. I. Akrotirianakis, I. Maros and B. Rustern (2001). An outer approximation based branch and cut algorithm for convex 0-1 MINLP problems. Optimization Methods and Software, 16, 21-47.
2. K.M. Anstreicher (1997). On Vaidya's volumetric cutting plane method for convex programming. Mathematics of Operations Research, 22, 63-89.
3. K.M. Anstreicher (1997). Volumetric path following algorithms for linear programming. Mathematical Programming, 76, 245-263.
4. K.M. Anstreicher (1998). The volumetric barrier for convex quadratic constraints. Technical Report, Department of Management Sciences, University of Iowa, Iowa City, Iowa 52242, October 1998.
5. K.M. Anstreicher (1999). Towards a practical volumetric cutting plane method for convex programming. SIAM Journal on Optimization, 9, 190-206.
6. K.M. Anstreicher (2000). The volumetric barrier for semidefinite programming. Mathematics of Operations Research, 25 (3), 365-380.
7. D.S. Atkinson and P.M. Vaidya (1995). A cutting plane algorithm for convex programming that uses analytic centers. Mathematical Programming, 69, 1-43.
8. O. Bahn, O. du Merle, J.L. Coffin and J.P. Vial (1995). A cutting plane method from analytic centers for stochastic programming. Mathematical Programming, 69, 45-73.
9. J.-L. Goffin, J. Gondzio, R. Sarkissian and J.-P. Vial (1997). Solving nonlinear multi-commodity network flow problems by the analytic center cutting plane method. Mathematical Programming, 76, 131-154.
10. J.-L. Goffin, Z.-Q. Luo and Y. Ye (1996). Complexity analysis of an interior cutting plane method for convex feasibility problems. SIAM Journal on Optimization, 6, 638-652.
11. J.-L. Coffin and J.-P. Vial (2001). Multiple cuts in the analytic center cutting plane method. SIAM Journal on Optimization, 11(1), 266-288.
12. J.-L. Goffin and J.-P. Vial (2002). Convex nondifferentiable optimization: A survey focussed on the analytic center cutting plane method. Optimization Methods and Software, 17(5), 805-867.
13. J. Gondzio (1998). Warm start of the primal - dual method applied in the cutting plane scheme. Mathematical Programming, 83, 125-143.
14. J. Gondzio, O. du Merle, R. Sarkissian and J.-P. Vial (1996). ACCPM - a library for convex optimization based on an analytic center cutting plane method, European Journal of Operational Research, 94, 206-211.
15. M. Grötschel, L. Lovasz and A. Schrijver (1988). Geometric Algorithms and Combinatorial Optimization. Springer-Verlag, Berlin, Germany.
16. D. den Hertog, J. Kaliski, C. Roos and T. Terlaky (1995). A logarithmic barrier cutting plane method for convex programming problems. Annals of Operations Research, 58, 69-98.
17. D. den Hertog, C. Roos and T. Terlaky (1992). A build-up variant of the path-following method for LP. Operations Research Letters, 12, 181-186.
18. D. den Hertog, C. Roos and T. Terlaky (1994). Adding and deleting constraints in the logarithmic barrier method for LP. In: D.-Z. Du and J. Sun (Eds.), Advances in Optimization and Approximation, pp. 166-185. Kluwer Academic Publishers, Dordrecht, The Netherlands.
19. J. Kaliski, D. Haglin, C. Roos and T. Terlaky (1997). Logarithmic barrier decomposition methods for semi-infinite programming. International Transactions in Operations Research, 4, 285-303.
20. K. Krishnan and J.E. Mitchell (2002). Cutting plane methods for semidefinite programming. Technical Report, Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY 12180, November.
21. K. Krishnan and J.E. Mitchell (2003). Semidefinite cutting plane approaches for the maxcut problem. Technical Report, Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY 12180, July.
22. Z.-Q. Luo, C. Roos and T. Terlaky (1999). Complexity analysis of a logarithmic barrier decomposition method for semi-infinite linear programming. Applied Numerical Mathematics, 29(3), 379-395. Special issue dedicated to the HPOPT-I workshop held in Delft, September 19-20, 1996.
23. N. Megiddo (1991). On finding primal- and dual-optimal bases. ORSA Journal on Computing, 3, 63-65.
24. J.E. Mitchell (1997). Fixing variables and generating classical cutting planes when using an interior point branch and cut method to solve integer programming problems. European Journal of Operational Research, 97, 139-148.
25. J.E. Mitchell (2000). Computational experience with an interior point cutting plane algorithm. SIAM Journal on Optimization, 10(4), 1212-1227.
26. J.E. Mitchell and B. Borchers (2000). Solving linear ordering problems with a combined interior point/simplex cutting plane algorithm. In : H.L. Frenk et al. (Ed.), High Performance Optimization, Chapter 14, pages 349-366. Kluwer Academic Publishers, Dordrecht, The Netherlands.
27. J.E. Mitchell, P.M. Pardalos and M.G.C. Resende (1998). Interior point methods for combinatorial optimization. In: D.-Z. Du and P.M. Pardalos (Eds.), Handbook of Combinatorial Optimization, Vol. 1, pp. 189-297. Kluwer Academic Publishers.
28. J.E. Mitchell and S. Ramaswamy (2000). A long-step, cutting plane algorithm for linear and convex programming. Annals of Operations Research, 99, 95-122.
29. J.E. Mitchell and M.J. Todd (1992). Solving combinatorial optimization problems using Karmarkar's algorithm. Mathematical Programming, 56, 245-284.
30. F.S. Mokhtarian and J.-L. Coffin (1998). A nonlinear analytic center cutting plane method for a class of convex programming problems. SIAM Journal on Optimization, 8, 1108-1131.
31. A.S. Nemirovsky and D.B. Yudin (1983). Problem Complexity and Method Efficiency in Optimization, John Wiley, New York, NY, USA.
32. Y.E. Nesterov and A.S. Nemirovsky (1993). Interior Point Polynomial Methods in Convex Programming: Theory and Algorithms. SIAM Publications, SIAM, Philadelphia, USA.
33. S. Ramaswamy and J.E. Mitchell (1995). A long step cutting plane algorithm that uses the volumetric barrier. Technical Report, DSES, Rensselaer Polytechnic Institute, Troy, NY 12180, June 1995.
34. J. Renegar (1988). A polynomial-time algorithm based on Newton's method for linear programming. Mathematical Programming, 40, 59-93.
35. P.M. Vaidya (1996). A new algorithm for minimizing convex functions over convex sets. Mathematical Programming, 73, 291-341.
36. Y. Ye, O. Güler, R.A. Tapia and Y. Zhang (1993). A quadratically convergent O(√nL)-iteration algorithm for linear programming. Mathematical Programming, 59, 151-162.
37. Y. Zhang, R.A. Tapia and J.E. Dennis (1992). On the superlinear and quadratic convergence of primal-dual interior point linear programming algorithms. SIAM Journal on Optimization, 2(2), 304-324.