Efficiency of ballstep subgradient level methods for convex optimization

Mathematics of Operations Research

Volumn 24, Issue 1, 1999, Pages 237-254

  Kiwiel, Krzysztof C ^a   Larsson, Torbjörn ^b   Lindberg, P O ^b  

^a SYSTEMS RESEARCH INSTITUTE   (Poland)

^b LINKÖPING UNIVERSITY   (Sweden)

Author keywords

[No Author keywords available]

Indexed keywords

APPROXIMATION THEORY; CONSTRAINT THEORY; CONVERGENCE OF NUMERICAL METHODS; OPTIMIZATION; SET THEORY;

BALLSTEP SUBGRADIENT LEVEL METHODS; CONVEX OPTIMIZATION;

OPERATIONS RESEARCH;

EID: 0033077459     PISSN: 0364765X     EISSN: None     Source Type: Journal    
DOI: 10.1287/moor.24.1.237     Document Type: Article

References (51)

1. Agmon, S. 1954. The relaxation method for linear inequalities. Canad. J. Math. 6 382-392.
2. Allen, E., R. Helgason, J. Kennington, B. Shetty. 1987. A generalization of Polyak's convergence result for subgradient optimization. Math. Programming 37 309-317.
3. Bazaraa, M. S., H. D. Sherali. 1981. On the choice of step size in subgradient optimization. European J. Oper. Res. 7 380-388.
4. Beasley, J. E. 1993. Lagrangean relaxation. C. R. Reeves, ed. Modern Heuristic Techniques for Combinatorial Problems. Blackwell Scientific Publications, Oxford, UK. 243-303.
5. Bertsekas, D. P. 1995. Nonlinear Programming. Athena Scientific, Belmont, MA.
6. Brännlund, U., K. C. Kiwiel, P. O. Lindberg. 1995. A descent proximal level bundle method for convex nondifferentiable optimization. Oper. Res. Lett. 17 121-126.
7. _. 1993. On Relaxation Methods for Nonsmooth Convex Optimization. Ph.D. Thesis, Department of Mathematics, Royal Institute of Technology, Stockholm, Sweden.
8. Bazaraa, M. S., H. D. Sherali, C. M. Shetty. 1993. Nonlinear Programming: Theory and Algorithms, 2nd ed. Wiley, New York.
9. Cegielski, A. 1994. Projection onto an acute cone and convex feasibility problem. J. Henry and J.-P. Yvon, eds. System Modelling and Optimization. Lecture Notes in Control and Information Sciences 197. Springer-Verlag, Berlin, Germany. 187-194.
10. Camerini, P. M., L. Fratta, F. Maffioli. 1975. On improving relaxation methods by modified gradient techniques. Math. Programming Stud. 3 26-34.
11. Demyanov, V. F., L. V. Vasilev. 1985. Nondifferentiable Optimization. Nauka, Moscow (Russian). English translation Optimization Software Inc., New York.
12. Drezner, Z. 1983. The nested ball principle for the relaxation method. Oper. Res. 31 587-590.
13. Ermoliev, Yu. M. 1966. Methods of solution of nonlinear extremal problems. Kibemetika no. 4 1-17 (Russian). English translation in Cybernetics 2 1-16.
14. Ermoliev, Yu. M. 1966. Methods of solution of nonlinear extremal problems. Kibemetika no. 4 1-17 (Russian). English translation in Cybernetics 2 1-16.
15. Fisher, M. L. 1985. An applications oriented guide to Lagrangian relaxation. Interfaces 15 10-21.
16. Goffin, J.-L. 1977. On convergence rates of subgradient optimization methods. Math. Programming 13 329-347.
17. _. 1981. Convergence results in a class of variable metric subgradient methods. O. L. Mangasarian, R. R. Meyer and S. M. Robinson, eds. Nonlinear Programming 4. Academic Press, New York. 283-326.
18. _, K. C. Kiwiel. 1998. Convergence of a simple subgradient level method. To appear in Math. Programming. Hiriart-Urruty, J.-B., C. Lemaréchal. 1993. Convex Analysis and Minimization Algorithms. Springer-Verlag, Berlin, Germany.
19. _, K. C. Kiwiel. 1998. Convergence of a simple subgradient level method. To appear in Math. Programming. Hiriart-Urruty, J.-B., C. Lemaréchal. 1993. Convex Analysis and Minimization Algorithms. Springer-Verlag, Berlin, Germany.
20. Held, M., P. Wolfe, H. P. Crowder. 1974. Validation of subgradient optimization. Math. Programming 6 62-88.
21. Johnson, D. S., L. A. McGeoch, E. E. Rothberg. 1996. Asymptotic experimental analysis for the Held-Karp traveling salesman bound. Proceedings of the Seventh Annual ACM-SlAM Symposium on Discrete Algorithms, SIAM, Philadelphia, PA. 341-350.
22. Johnson, D. S. 1998. Private communication.
23. Kim, S., H. Ahn, S.-C. Cho. 1991. Variable target value subgradient method. Math. Programming 49 359-369.
24. _, B.-S. Um. 1993. An improved subgradient method for constrained nondifferentiable minimization. Oper. Res. Lett. 14 61-64.
25. Kiwiel, K. C. 1990. Proximity control in bundle methods for convex nondifferentiable minimization. Math. Programming 46 105-122.
26. _. 1994. A Cholesky dual method for proximal piecewise linear programming. Numer. Math. 68 325-340.
27. _. 1995a. Block-iterative surrogate projection methods for convex feasibility problems. Linear Algebra Appl. 215 225-260.
28. _, 1995b. Finding normal solutions in piecewise linear programming. Appl. Math. Optim. 32 235-254.
29. _. 1995c. Proximal level bundle methods for convex nondifferentiable optimization, saddle-point problems and variational inequalities. Math. Programming 69 89-109.
30. _. 1996a. The efficiency of subgradient projection methods for convex optimization, Part I: General level methods. SIAM J. Control Optim. 34 660-676.
31. _ 1996b. The efficiency of subgradient projection methods for convex optimization, Part II: Implementations and extensions. SIAM J. Control Optim. 34 677-697.
32. _. 1997. Monotone Gram matrices and deepest surrogate inequalities in accelerated relaxation methods for convex feasibility problems. Linear Algebra Appl. 252 27-33.
33. Kulikov, A. N., V. R. Fazilov. 1990. Convex optimization with prescribed accuracy. Zh. Vychisl. Mat. i Mat. Fiz. 30 663-671 (Russian).
34. Lemaréchal, C., A. S. Nemirovskii, Yu. E. Nesterov. 1995. New variants of bundle methods. Math. Programming 69 111-147.
35. Marsten, R. E., W. W. Hogan, J. W. Blankenship. 1975. The Boxstep method for large scale optimization. Oper. Res. 23 389-405.
36. Minoux, M. 1986. Mathematical Programming, Theory and Algorithms. Wiley, Chichester, UK.
37. Nesterov, Yu. E. 1984. Minimization methods for nonsmooth convex and quasiconvex functions. Èkonom. i Mat, Metody 20 519-531 (Russian). English translation in Matekon 29 (3) 519-531.
38. Nesterov, Yu. E. 1984. Minimization methods for nonsmooth convex and quasiconvex functions. Èkonom. i Mat, Metody 20 519-531 (Russian). English translation in Matekon 29 (3) 519-531.
39. _. 1989. Effective Methods in Nonlinear Programming. Radio i Sviaz, Moscow (Russian).
40. Polyak, B. T. 1967. A general method for solving extremum problems. Dokl. Akad. Nauk SSSR 174 33-36 (Russian). English translation in Soviet Math. Dokl. 8 593-597.
41. Polyak, B. T. 1967. A general method for solving extremum problems. Dokl. Akad. Nauk SSSR 174 33-36 (Russian). English translation in Soviet Math. Dokl. 8 593-597.
42. _. 1969. Minimization of unsmooth functionals. Zh. Vychisl. Mat. i Mat. Fiz. 9 509-521 (Russian). English translation in U.S.S.R. Comput. Math, and Math. Phys. 9 14-29.
43. _. 1969. Minimization of unsmooth functionals. Zh. Vychisl. Mat. i Mat. Fiz. 9 509-521 (Russian). English translation in U.S.S.R. Comput. Math, and Math. Phys. 9 14-29.
44. Reinelt, G. 1995. TSPLIB 95, Tech. Report, Institut für Angewandte Mathematik, Universität Heidelberg, Heidelberg, Germany, ftp://ftp.iwr.uni-heidelberg.de/pub/tsplib.
45. Rockafellar, R. T. 1970. Convex Analysis. Princeton University Press, Princeton, NJ.
46. Shchepakin, M. B. 1992. An algorithm of orthogonal descent for searching for the zero of a convex function, identification of unsolvability of the problem and the questions of the speed of convergence. Kibernetika no. 5 87-96 (Russian).
47. Shor, N. Z. 1979. Minimization Methods for Non-Differentiable Functions. Naukova Dumka, Kiev (Russian). English translation, Springer-Verlag, Berlin, Germany. 1985.
48. Sherali, H. D., O. Ulular. 1989. A primal-dual conjugate subgradient algorithm for specially structured linear and convex programming problems. Appl. Math. Optim. 20 193-221.
49. Todd, M. J. 1979. Some remarks on the relaxation method for linear inequalities. Tech. Report 419, Cornell Univ., Cornell, Ithaca, NY.
50. Valenzuela, C. L., A. J. Jones. 1997. Estimating the Held-Karp lower bound for the geometric TSP. European J. Oper. Res. 102 157-175.
51. Volgenant, T., R. Jonker. 1982. A branch and bound algorithm for the symmetric traveling salesman problem based on the 1-tree relaxation. European J. Oper. Res. 9 83-89.