Cost approximation: A unified framework of descent algorithms for nonlinear programs

SIAM Journal on Optimization

Volumn 8, Issue 2, 1998, Pages 561-582

  Patriksson, Michael ^a  

^a CHALMERS UNIVERSITY OF TECHNOLOGY   (Sweden)

Author keywords

Convergence analysis; Cost approximation; Descent algorithms; Nondifferentiable optimization; Partial linearization

Indexed keywords

EID: 0032222093     PISSN: 10526234     EISSN: None     Source Type: Journal    
DOI: 10.1137/S105262349427577X     Document Type: Article

References (52)

1. L. ARMIJO, Minimization of functions having Lipschitz continuous first partial derivatives, Pacific J. Math., 16 (1966), pp. 1-3.
2. C. BERGE, Topological Spaces, Oliver & Boyd, Edinburgh, 1963.
3. H. BRÉZIS, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, North-Holland, Amsterdam, 1973.
4. F. E. BROWDER, Nonlinear maximal monotone mappings in Banach spaces, Math. Ann., 175 (1968), pp. 81-113.
5. G. H.-G. CHEN AND R. T. ROCKAFELLAR, Forward-Backward Splitting Methods in Lagrangian Optimization, Report, Department of Applied Mathematics, University of Washington, Seattle, WA, 1992.
6. G. COHEN, Optimization by decomposition and coordination: A unified approach, IEEE Trans. Automat. Control, AC-23 (1978), pp. 222-232.
7. G. COHEN, Auxiliary problem principle and decomposition of optimization problems, J. Optim. Theory Appl., 32 (1980), pp. 277-305.
8. G. COHEN AND D. L. ZHU, Decomposition coordination methods in large scale optimization problems: The nondifferentiable case and the use of augmented Lagrangians, in Advances in Large Scale Systems, Vol. 1, J. B. Cruz, ed., JAI Press, Greenwich, CT, 1984, pp. 203-266.
9. R. W. COTTLE, J.-S. PANG, AND R. E. STONE, The Linear Complementarity Problem, Academic Press, San Diego, CA, 1992.
10. V. F. DEM'YANOV AND L. V. VASIL'EV, Nondifferentiable Optimization, Optimization Software, New York, 1985.
11. J.-P. DUSSAULT, J. A. FERLAND, AND B. LEMAIRE, Convex quadratic programming with one constraint and bounded variables, Math. Programming, 36 (1986), pp. 90-104.
12. J. ECKSTEIN AND D. P. BERTSEKAS, On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators, Math. Programming, 55 (1992), pp. 293-318.
13. M. FUKUSHIMA, M. HADDOU, V. H. NGUYEN, J.-J. STRODIOT, T. SUGIMOTO, AND E. YAMAKAWA, A parallel descent algorithm for convex programming, Comput. Optim. Appl., 5 (1996), pp. 5-37.
14. M. FUKUSHIMA AND H. MINE, A generalized proximal point algorithm for certain nonconvex minimization problems, Internat. J. Systems Sci., 12 (1981), pp. 989-1000.
15. A. A. GOLDSTEIN, Convex programming in Hilbert space, Bull. Amer. Math. Soc., 70 (1964), pp. 709-710.
16. T. LARSSON AND A. MIGDALAS, An algorithm for nonlinear programs over Cartesian product sets, Optimization, 21 (1990), pp. 535-542.
17. T. LARSSON AND M. PATRIKSSON, A class of gap functions for variational inequalities, Math. Programming, 64 (1994), pp. 53-79.
18. L. J. LEBLANC, R. V. HELGASON, AND D. E. BOYCE, Improved efficiency of the Frank-Wolfe algorithm for convex network programs, Transportation Sci., 19 (1985), pp. 445-462.
19. E. S. LEVITIN AND B. T. POLYAK, Constrained minimization methods, U.S.S.R. Comput. Math, and Math. Phys., 6 (1966), pp. 1-50.
20. D. G. LUENBERGER, Linear and Nonlinear Programming, 2nd ed., Addison-Wesley, Reading, MA, 1984.
21. B. MARTINET, Regularisation d'inéquations variationnelles par approximations successives, Revue Française d'Informatique et de Recherche Operationelle, R-3 (1970), pp. 154-158.
22. G. G. L. MEYER, Asymptotic properties of sequences iteratively generated by point-to-set maps, Math. Programming Study, 10 (1979), pp. 115-127.
23. A. MIGDALAS, A regularization of the Frank-Wolfe method and unification of certain nonlinear programming methods, Math. Programming, 65 (1994), pp. 331-345.
24. H. MINE AND M. FUKUSHIMA, A minimization method for the sum of a convex function and a continuously differentiable function, J. Optim. Theory Appl., 33 (1981), pp. 9-23.
25. G. J. MINTY, Monotone (nonlinear) operators in Hilbert space, Duke Math. J., 29 (1962), pp. 341-346.
26. H. NIKAIDO, Convex Structures and Economic Theory, Academic Press, New York, 1968.
27. J. M. ORTEGA AND W. C. RHEINBOLDT, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970.
28. M. PATRIKSSON, Partial linearization methods in nonlinear programming, J. Optim. Theory Appl., 78 (1993), pp. 227-246.
29. M. PATRIKSSON, A unified description of iterative algorithms for traffic equilibria, European J. Oper. Res., 71 (1993), pp. 154-176.
30. M. PATRIKSSON, A unified framework of descent algorithms for nonlinear programs and variational inequalities, Ph.D. dissertation, Department of Mathematics, Linköping Institute of Technology, Linköping, Sweden, 1993.
31. M. PATRIKSSON, On the convergence of descent methods for monotone variational inequalities, Oper. Res. Lett., 16 (1994), pp. 265-269.
32. M. PATRIKSSON, The Traffic Assignment Problem: Models and Methods, VSP, Utrecht, 1994.
33. M. PATRIKSSON, Merit functions and descent algorithms for a class of variational inequality problems, Optimization, 41 (1997), pp. 37-55.
34. M. PATRIKSSON, Nonlinear Programming and Variational Inequalities: A Unified Approach, Kluwer Academic Publishers, Norwell, MA, to appear.
35. B. T. POLYAK, Gradient methods for the minimisation of functional, U.S.S.R. Comput. Math. and Math. Phys., 3 (1963), pp. 864-878.
36. B. T. POLYAK, Introduction to Optimization, Optimization Software, New York, 1987.
37. B. N. PSHENICHNY AND YU. M. DANILIN, Numerical Methods in Extremal Problems, Mir Publishers, Moscow, 1978.
38. S. M. ROBINSON, Generalized equations and their solutions, part I: Basic theory, Math. Programming Study, 10 (1979), pp. 128-141.
39. R. T. ROCKAFELLAR, Characterization of the subdifferentials of convex functions, Pacific J. Math., 17 (1966), pp. 497-510.
40. R. T. ROCKAFELLAR, Local boundedness of nonlinear, monotone operators, Michigan Math. J., 16 (1969), pp. 397-407.
41. R. T. ROCKAFELLAR, Convex Analysis, Princeton University Press, Princeton, NJ, 1970.
42. R. T. ROCKAFELLAR, On the maximal monotonicity of subdifferential mappings, Pacific J. Math., 33 (1970), pp. 209-216.
43. R. T. ROCKAFELLAR, On the maximality of sums of nonlinear monotone operators, Trans. Amer. Math. Soc., 149 (1970), pp. 75-88.
44. R. T. ROCKAFELLAR, Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14 (1976), pp. 877-898.
45. R. T. ROCKAFELLAR, The Theory of Subgradients and its Applications to Problems of Optimization: Convex and Nonconvex Functions, Heldermann-Verlag, Berlin, 1981.
46. N. Z. SHOR, Minimization Methods for Non-Differentiable Functions, Springer-Verlag, Berlin, 1985.
47. J. STOER AND C. WITZGALL, Convexity and Optimization in Finite Dimensions I, Springer-Verlag, Berlin, 1970.
48. P. TSENG, Decomposition algorithm for convex differentiable minimization, J. Optim. Theory Appl., 70 (1991), pp. 109-135.
49. G. R. WOOD AND B. P. ZHANG, Estimation of the Lipschitz constant of a function, J. Global Optim., 8 (1996), pp. 91-103.
50. W. I. ZANGWILL, Nonlinear Programming: A Unified Approach, Prentice-Hall Englewood Cliffs, NJ, 1969.
51. E. ZEIDLER, Nonlinear Functional Analysis and Its Applications II/B: Nonlinear Monotone Operators, Springer-Verlag, New York, 1990.
52. D. ZHU AND P. MARCOTTE, Coupling the auxiliary problem principle with descent methods of pseudoconvex programming, European J. Oper. Res., 83 (1995), pp. 670-685.