Efficiency of the analytic center cutting plane method for convex minimization

SIAM Journal on Optimization

Volumn 7, Issue 2, 1997, Pages 336-346

  Kiwiel, Krzysztof C ^a  

^a SYSTEMS RESEARCH INSTITUTE   (Poland)

Author keywords

Analytic center; Cutting plane methods; Nondifferentiable optimization; Potential function

Indexed keywords

EID: 0031531864     PISSN: 10526234     EISSN: None     Source Type: Journal    
DOI: 10.1137/S1052623494275768     Document Type: Article

References (28)

1. A. ALTMAN AND K. C. KIWIEL, A note on some analytic center cutting plane methods for convex feasibility and minimization problems, Comput. Optim. Appl., 5 (1996), pp. 175-180.
2. A. ALTMAN, Generalized Karmarkar Proactive Methods for Convex Nondifferentiable Optimization Problems, Ph.D. thesis, Systems Research Institute, Warsaw, Poland, 1994 (in Polish).
3. K. M. ANSTREICHER, On Vaidya's Volumetric Cutting Plane Method for Convex Programming, Tech. report, Department of Management Sciences, University of Iowa, Iowa City, IA, September, 1994.
4. D. S. ATKINSON AND P. M. VAIDYA, A scaling technique for finding the weighted analytic center of a polytope, Math. Programming, 57 (1992), pp. 163-192.
5. D. S. ATKINSON AND P. M. VAIDYA, A cutting plane algorithm for convex programming that uses analytic centers, Math. Programming, 69 (1995), pp. 1-43.
6. O. BAHN. O. DU MERLE, J.-L. COFFIN, AND J.-PH. VIAL, A cutting plane method from analytic centers for stochastic programming, Math. Programming, 69 (1995), pp. 45-73.
7. O. BAHN, J.-L. GOFFIN, J.-PH. VIAL, AND O. DU MERLE, Implementation and behavior of an interior point cutting plane algorithm for convex programming: An application to geometric programming, Discrete Appl. Math., 49 (1993), pp. 3-23.
8. D. DEN HERTOG, C. ROOS, AND T. TERLAKY, Adding and deleting constraints in the logarithmic barrier method for linear programming, in Advances in Optimization and Approximation, D.-Z. Du and J. Sun, eds., Nonconvex Optimization and Applications, Kluwer, Dordrecht, 1994, pp. 166-185.
9. A. V. FIACCO AND G. P. MCCORMICK, Nonlinear Programming: Sequential Unconstrained Minimization Techniques, Wiley, New York, 1968.
10. J.-L. GOFFIN, J. GONDZIO, R. SARKISSIAN, AND J.-PH. VIAL, Solving nonlinear multicommodity flow problems by the analytic center cutting plane method, Tech. report, Département d'économie commerciale et industrielle, Université de Genève, Genève, Switzerland, October, 1994.
11. J.-L. COFFIN, A. HAURIE, AND J.-PH. VIAL, Decomposition and nondifferentiable optimization with the protective algorithm, Management Sci., 37 (1992), pp. 284-302.
12. J.-L. COFFIN, Z.-Q. LUO, AND Y. YE, On the complexity of a column generation algorithm for convex or quasiconvex feasibility problems, in Large Scale Optimization: State of the Art, W. W. Hager, D. W. Hearn, and P. M. Pardalos, eds., Kluwer, Dordrecht, 1994, pp. 182-191.
13. J.-L. GOFFIN, Z.-Q. LUO, AND Y. YE, Complexity analysis of an interior point cutting plane method for convex feasibility problems, SIAM J. Optim., (1996), pp. 638-652.
14. J.-L. GOFFIN, Using the Primal Dual Infeasible Newton Method in the Analytic Center Method for Problems Defined by Deep Cutting Planes, Tech. report, Faculty of Management, McGill University, Montreal, Quebec, September, 1994.
15. J.-L. GOFFIN AND J.-PH. VIAL, On the computation of weighted analytic centers and dual ellipsoids with the protective algorithm, Math. Programming, 60 (1993), pp. 81-92.
16. K. C. KIWIEL, Complexity of some cutting plane methods that use analytic centers, Math. Programming, 74 (1996), pp. 47-54.
17. Z.-Q. LUO, Analysis of a Cutting Plane Method that Uses Analytic Center and Multiple Cuts, Tech. report, Department of Electrical and Computer Engineering, McMaster University, Hamilton, Ontario, Canada, September, 1994.
18. J. E. MITCHELL AND S. RAMASWAMY, A long-step, cutting plane algorithm for linear and convex programming, DSES Tech. report 37-93-387, Department of Decision Sciences & Engineering Systems, Rensselaer Polytechnic Institute, Troy, NY, August, 1993 (revised August, 1994).
19. YU. NESTEROV AND A. NEMIROVSKII, Interior-Point Polynomial Algorithms in Convex Programming, Studies in Applied Mathematics 13, SIAM, Philadelphia, PA, 1994.
20. YU. E. NESTEROV, Complexity estimates of some cutting plane methods based on the analytic barrier, Math. Programming, 69 (1995), pp. 149-176.
21. A. S. NEMIROVSKII AND D. B. YUDIN, Problem Complexity and Method Efficiency in Optimization, Nauka, Moscow, 1979 (in Russian).
22. S. RAMASWAMY AND J. E. MITCHELL, On updating the analytic center after the addition of multiple cuts, DSES Tech. report 37-94-423, Department of Decision Sciences & Engineering Systems, Rensselaer Polytechnic Institute, Troy, NY, October, 1994.
23. J. RENEGAR, A polynomial-time algorithm, based on Newton's method, for linear programming, Math. Programming, 40 (1988), pp. 59-93.
24. G. SONNEVEND, New algorithms in convex programming based on the notion of "centre" (for systems of analytic inequalities) and on rational extrapolation, in Trends in Mathematical Optimization, K. H. Hoffman, J.-B. Hiriart-Urruty, C. Lemaréchal, and J. Zowe, eds., International Series of Numerical Mathematics 84, Birkhäuser-Verlag, Basel, Switzerland, 1988, pp. 311-326.
25. 1.5n)L) arithmetic operations, Math. Programming, 47 (1990), pp. 175-201.
26. P. M. VAIDYA, A new algorithm for minimizing convex functions over convex sets, Math. Programming, 73 (1996), pp. 291-341.
27. Y. YE, A potential reduction algorithm allowing column generation, SIAM J. Optim., 2 (1992), pp. 7-20.
28. Y. YE, Complexity Analysis of the Analytic Center Cutting Plane Method that Uses Multiple Cuts, Tech. report, Department of Management Sciences, University of Iowa, Iowa City, IA, September, 1994.