Dual applications of proximal bundle methods, including Lagrangian relaxation of nonconvex problems

SIAM Journal on Optimization

Volumn 10, Issue 3, 2000, Pages 697-721

  Feltenmark, Stefan ^a   Kiwiel, Krzysztof C ^b  

^a ROYAL INSTITUTE OF TECHNOLOGY   (Sweden)

^b SYSTEMS RESEARCH INSTITUTE   (Poland)

Author keywords

Convex programming; Convexified relaxations; Lagrangian relaxation; Nondifferentiable optimization; Proximal bundle methods; Unit commitment

Indexed keywords

EID: 0034377204     PISSN: 10526234     EISSN: None     Source Type: Journal    
DOI: 10.1137/S1052623498332336     Document Type: Article

References (52)

1. K. M. ANSTREICHER AND L. WOLSEY, On Dual Solutions in Subgradient Optimization, Tech. report, Dept. of Management Sciences, University of Iowa, Iowa City, IA, September 1993.
2. J. F. BARD, Short-term scheduling of thermal-electric generators using Lagrangian relaxation, Oper. Res., 36 (1988), pp. 756-766.
3. D. P. BERTSEKAS, Constrained Optimization and Lagrange Multiplier Methods, Academic Press, New York, 1982.
4. D. P. BERTSEKAS, Nonlinear Programming, Athena Scientific, Belmont, MA, 1995.
5. D. BERTSEKAS, G. S. LAUER, N. R. SANDELL, AND T. A. POSBERG, Optimal shortterm scheduling of large-scale power systems, IEEE Trans. Automat. Control, AC-28 (1983), pp. 1-11.
6. J. V. BURKE AND M. C. FERRIS, Characterization of solutions sets of convex programs, Oper. Res. Lett., 10 (1991), pp. 57-60.
7. C. H. CHEUNG AND M. J. H. STERLING, Large-scale unit commitment using a composite thermal generator operator cost function, in Proceedings of the Second International Conference on Power Systems Monitoring and Control, IEEE Press, Piscataway, 1986, pp. 332-337.
8. R. CORREA AND C. LEMARÉCHAL, Convergence of some algorithms for convex minimization, Math. Programming, 62 (1993), pp. 261-275.
9. G. B. DANTZIG, Linear Programming and Extensions, Princeton University Press, Princeton, NJ, 1963.
10. D. DENTCHEVA, R. GOLLMER, A. MÖLLER, W. RÖMISCH, AND R. SCHULTZ, Solving the unit commitment problem in power generation by primal and dual methods, in Progress in Industrial Mathematics at ECMI 96, M. Brøns, M. P. Bendsøe, and M. P. Sørensen, eds., Teubner, Stuttgart, 1997, pp. 332-339.
11. B. FARDANESH AND F. E. VILLASECA, Two-step optimal thermal generation scheduling, Automatica, 22 (1986), pp. 361-366.
12. S. FELTENMARK AND K. C. KIWIEL, Generalized Linear Programming Solves the Relaxed Primal, Tech. report TRITA/MAT-97-OS11, Dept. of Mathematics, Royal Institute of Technology, Stockholm, Sweden, August 1997.
13. S. FELTENMARK, On Optimization of Power Production, Ph.D. thesis, Dept. of Mathematics, Royal Institute of Technology, Stockholm, Sweden, 1997.
14. M. L. FISHER, The Lagrangian relaxation method for solving integer programming problems, Management Sci., 27 (1981), pp. 1-18.
15. J.-B. HIRIART-URRUTY AND C. LEMARÉCHAL, Convex Analysis and Minimization Algorithms, Springer-Verlag, Berlin, 1993.
16. S. A. KAZARLIS, A. G. BAKIRTZIS, AND V. PETRIDIS, A genetic algorithm solution to the unit commitment problem, IEEE Trans. Power Systems, 11 (1996), pp. 83-91.
17. K. C. KIWIEL, A dual method for certain positive semidefimte quadratic programming problems, SIAM J. Sci. Statist. Comput., 10 (1989), pp. 175-186.
18. K. C. KIWIEL, Proximity control in bundle methods for convex nondifferentiable minimization, Math. Programming, 46 (1990), pp. 105-122.
19. K. C. KIWIEL, Exact penalty functions in proximal bundle methods for constrained convex nondifferentiable minimization, Math. Programming, 52 (1991), pp. 285-302.
20. K. C. KIWIEL, A Cholesky dual method for proximal piecewise linear programming, Numer. Math., 68 (1994), pp. 325-340.
21. K. C. KIWIEL, Approximations in proximal bundle methods and decomposition of convex programs, J. Optim. Theory Appl., 84 (1995), pp. 529-548.
22. K. C. KIWIEL, Restricted step and Levenberg-Marquardt techniques in proximal bundle methods for nonconvex nondifferentiable optimization, SIAM J. Optim., 6 (1996), pp. 227-249.
23. T. LARSSON AND Z. LIU, A Primal Convergence Result for Dual Subgradient Optimization with Application to Multicommodity Network Flows, Tech. report, Dept. of Mathematics, Linköping University, Linköping, Sweden, 1989.
24. S. LASDON, Optimization Theory for Large Systems, MacMillan, New York, 1970.
25. C. LEMARÉCHAL, An extension of Davidon methods to nondifferentiable problems. Non-differentiate optimization, Math. Programming Stud., 3 (1975), pp. 95-109.
26. C. LEMARÉCHAL, Nonsmooth Optimization and Descent Methods, Research report RR-78-4, International Institute of Applied Systems Analysis, Laxenburg, Austria, 1977.
27. C. LEMARÉCHAL AND A. RENAUD, Dual-equivalent Convex and Nonconvex Problems, Research report, INRIA, Rocquencourt, Prance, 1996.
28. C. LEMARÉCHAL AND C. SAGASTIZÁBAL, Variable metric bundle methods: From conceptual to implementable forms, Math. Programming, 76 (1997), pp. 393-410.
29. C. A. LI, R. A. JOHNSON, AND A. J. SVOBODA, A new unit commitment method, IEEE Trans. Power Systems, 12 (1997), pp. 113-119.
30. C. LEMARÉCHAL, F. PELLEGRINO, A. RENAUD, AND C. SAGASTIZÁBAL, Bundle methods applied to the unit-commitment problem, in System Modelling and Optimization, Proceedings of the Seventeenth IFIP TC7 Conference on System Modelling and Optimization, 1995, J. Doležal and J. Fidler, eds., Chapman & Hall, London, 1996, pp. 395-402.
31. T. LARSSON, M. PATRIKSSON, AND A.-B. STRÖMBERG, Ergodic convergence in subgradient optimization, Optim. Methods Softw., 9 (1998), pp. 93-120.
32. T. LARSSON, M. PATRIKSSON, AND A.-B. STRÖMBERG, Ergodic, primal convergence in dual subgradient schemes for convex programming, Math. Programming, 86 (1999), pp. 283-312.
33. C. LEMARÉCHAL, J.-J. STRODIOT, AND A. BIHAIN, On a bundle algorithm for nonsmooth optimization, in Nonlinear Programming 4, O. L. Mangasarian, R. R. Meyer, and S. M. Robinson, eds., Academic Press, New York, 1981, pp. 245-282.
34. R. MIFFLIN, A modification and an extension of Lemaréchal's algorithm for nonsmooth minimization, Math. Programming Stud., 17 (1982), pp. 77-90.
35. T. L. MAGNANTI, J. F. SHAPIRO, AND M. H. WAGNER, Generalized linear programming solves the dual, Management Sci., 23 (1976), pp. 1195-1203.
36. J. A. MUCKSTADT AND S. A. KOENIG, An application of Lagrangian relaxation to scheduling in thermal power-generation systems, Oper. Res., 25 (1977), pp. 387-403.
37. I. RISH, private communication, Department of Information and Computer Science, University of California, Irvine, CA, September, 1996.
38. S. M. ROBINSON, Bundle-based decomposition: Description and preliminary results, in System Modelling and Optimization, Lecture Notes in Control and Inform. Sci. 84, A. Prékopa, J. Szelezsán, and B. Strazicky, eds., Springer-Verlag, Berlin, 1986, pp. 751-756.
39. S. M. ROBINSON, Bundle-based decomposition: Conditions for convergence, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), pp. 435-447.
40. S. V. RZHEVSKII, A conditional ∈-subgradient method for the simultaneous solution of the dual and primal problems of convex programming, Kibernetika, (1989), pp. 54-64 (in Russian). English translation in Cybernetics, 25 (1989), pp. 203-218.
41. S. V. RZHEVSKII, A conditional ∈-subgradient method for the simultaneous solution of the dual and primal problems of convex programming, Kibernetika, (1989), pp. 54-64 (in Russian). English translation in Cybernetics, 25 (1989), pp. 203-218.
42. S. V. RZHEVSKII AND A. V. KUNCEVICH, An application of the ∈-subgradient method to the solution of the dual and primal problems of convex programming, Kibernetika, (1985), pp. 51-54 (in Russian).
43. H. SCHRAMM AND J. ZOWE, A combination of the bundle approach and the trust region concept, in Advances in Mathematical Optimization, Math. Res. 45, J. Guddat et al., eds., Akademie-Verlag, Berlin, 1988, pp. 196-209.
44. J. F. SHAPIRO, Mathematical Programming: Structures and Algorithms, Wiley, New York, 1979.
45. J. J. SHAW, A direct method for security-constrained unit commitment, IEEE Trans. Power Systems, 10 (1995), pp. 1329-1342.
46. H. D. SHERALI AND G. CHOI, Recovery of primal solutions when using subgradient optimization methods to solve Lagrangian duals of linear programs, Oper. Res. Lett., 19 (1996), pp. 105-113.
47. G. B. SHEBLÉ AND G. N. FAHD, Unit commitment literature synopsis, IEEE Trans. Power Systems, 9 (1994), pp. 128-135.
48. N. Z. SHOR, Minimization Methods for Non-Differentiable Functions, Naukova Dumka, Kiev, 1979 (in Russian). English translation published by Springer-Verlag, Berlin, 1985.
49. P. WOLFE, A method of conjugate subgradients for minimizing nondifferentiable functions, Math. Programming Stud., 3 (1975), pp. 145-173.
50. G. ZHUANG AND F. D. GALIANA, Towards a more rigorous and practical unit commitment by Lagrangian relaxation, IEEE Trans. Power Systems, 3 (1988), pp. 763-773.
51. N. G. ZHURBENKO, Investigation of a Class of Algorithms for Minimizing Nonsmooth Functions and Their Application to the Solution of Large-Scale Problems, Dissertation, Cybernetics Institute, Academy of Sciences of the Ukrainian SSR, Kiev, 1977 (in Russian).
52. H. W. ZEMINGER, A. J. WOOD, H. K. CLARK, T. F. LASKOWSKI, AND J. D. BURNS, Synthetic Electric Utility Systems for Evaluating Advanced Technologies, Final report EM-285, Electrical Power Research Institute (EPRI), Palo Alto, CA, 1977.