Abstract
In this paper, we consider the nonlinear programming problem P to minimize f(x) subject to g i (x)<=0 for i=1, …, m and x∈X. If X is compact and the number of global optimal solutions is finite, under a suitable constraint qualification, we show that a globally exact penalty function exists. Particularly, we establish a one to one correspondence between global optimal solutions to the original problem and global minimizers of the penalty problem for a sufficiently large, but finite, penalty parameter. A lower bound on the penalty parameter is provided in terms of the Kuhn-Tucker Lagrangian multipliers and lower bounds on the functions involved.
This author’s work is supported under USAFOSR contract F49630-79-C0120.
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References
M.S. Bazaraa and J.J. Goode, “Necessary optimality criteria in mathematical programming in the presence of differentiability”, Journal of Mathematical Analysis and Applications 40 (1972) 609–621.
M.S. Bararaa and C.M. Shetty, Nonlinear programming: Theory and Algorithms (Wiley, New York, 1979).
D.P. Bertsekas, “Necessary and sufficient conditions for a penalty method to be exact”, Mathematical Programming 9 (1975) 87–99.
C. Charalambous, “A lower bound for the controlling parameter of the exact penalty functions”, Mathematical Programming 15 (1978) 278–290.
J.P. Evans, F.J. Gould and J.W. Tolle, “Exact penalty functions in nonlinear programming”, Mathematical Programming 4 (1973) 72–97.
R. Fletcher, “An exact penalty function for nonlinear programming with inequalities”, Mathematical Programming 5 (1973) 129–150.
M. Guignard, “Generalized Kuhn-Tucker conditions for mathematical programming problems in a Banach space”, SIAM Journal on Control 7 (1969) 232–241.
S.P. Han, “A globally convergent method for nonlinear programming”, Journal of Optimization Theory and Applications 22 (1977) 297–309.
S.P. Han and O.L. Mangasarian, “Exact penalty functions in nonlinear programming”, Mathematical Programming 17 (1979) 251–269.
S. Howe, “New conditions for exactness of a simple penalty function”, SIAM Journal on Control 11 (1973) 378–381.
O.L. Mangasarian, Nonlinear programming (McGraw-Hill, New York, 1969).
G.P. McCormick, “An idealized exact penalty function”, in: O.L. Mangasarian, R.R. Meyer and S.M. Robinson, eds., Nonlinear programming, Vol. 3 (Academic Press, New York, 1978) pp. 165–195.
T. Pietrzykowski, “The exact potential method for constrained maxima”, SIAM Journal on Numerical Analysis 6 (1969) 299–304.
T. Pietrzykowski, “The potential method for conditioned maxima in locally compact metric spaces”, Numerische Mathematik 14 (1970) 325–329.
R.T. Rockafellar, “Theory of subgradients and its applications to problems of optimization”, lecture notes, University of Montreal (1978).
W.I. Zangwill, “Nonlinear programming via penalty functions”, Management Science 13 (1967) 344–358.
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© 1982 The Mathematical Programming Society, Inc.
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Bazaraa, M.S., Goode, J.J. (1982). Sufficient conditions for a globally exact penalty function without convexity. In: Guignard, M. (eds) Optimality and Stability in Mathematical Programming. Mathematical Programming Studies, vol 19. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0120980
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DOI: https://doi.org/10.1007/BFb0120980
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Publisher Name: Springer, Berlin, Heidelberg
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