Abstract
The nonlinear parametric programming problem is reformulated as a closed system of nonlinear equations so that numerical continuation and bifurcation techniques can be used to investigate the dependence of the optimal solution on the system parameters. This system, which is motivated by the Fritz John first-order necessary conditions, contains all Fritz John and all Karush-Kuhn-Tucker points as well as local minima and maxima, saddle points, feasible and nonfeasible critical points. Necessary and sufficient conditions for a singularity to occur in this system are characterized in terms of the loss of a complementarity condition, the linear dependence of the gradients of the active constraints, and the singularity of the Hessian of the Lagrangian on a tangent space. Any singularity can be placed in one of seven distinct classes depending upon which subset of these three conditions hold true at a solution. For problems with one parameter, we analyze simple and multiple bifurcation of critical points from a singularity arising from the loss of the complementarity condition, and then develop a set of conditions which guarantees the unique persistence of a minimum through this singularity.
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References
E. L. Allgower and K. Georg, “Predictor-Corrector and simplical methods for approximating fixed points and zero points of nonlienar mappings,” in: A. Bachem, M. Grötschel and B. Korte, eds.,Mathematical Programming The State of the Art (Springer-Verlag, New York, 1983) pp. 15–36.
B. Bank, J. Guddat, D. Klatte, B. Kummer and K. Tammer,Nonlinear Parametric Optimization (Birkhauser-Verlag, Basel, 1983).
S.-N. Chow and J.K. Hale,Methods of Bifurcation Theory (Springer-Verlag, New York, 1982).
E.J. Doedel, “AUTO: A program for the automatic bifurcation analysis of autonomous systems,” Cong. Num. 30 (1981) 265–284,Proceedings of the Tenth Manitoba Conference on Numerical Mathematics and Computing (Winnipeg, Canada).
E.J. Doedel, A.D. Jepson and H.B. Keller, “Numerical methods for Hopf bifurcation and continuation of periodic solution paths,” in: R. Glowinski and J.L. Lions, eds.,Computing Methods in Applied Sciences and Engineering VI (North-Holland, Amsterdam, 1984) pp 127–138.
A.V. Fiacco,Introduction to Sensitivity and Stability Analysis in Nonlinear Programming (New York, Academic Press, 1983).
A.V. Fiacco,Mathematical Programming Study 21: Sensitivity, Stability and Parametric Analysis (North-Holland, Amsterdam, 1984).
R. Fletcher,Practical Methods of Optimization Vol. 2 (John Wiley and Sons, New York, 1981).
P.E. Gill, W. Murray, M. Wright,Practical Optimization (Academic Press, London, 1981).
G.H. Golub and C.F. Van Loan,Matrix Computations (The Johns Hopkins University Press, Baltimore, 1983).
M. Golubitsky and D.G. Schaeffer,Singularities and Groups in Bifurcation Theory, Vol. 1 (Springer-Verlag, New York, 1985).
C.D. Ha, “Application of degree theory in stability of the complementarity problem,” to appear inMathematics of Operations Research.
G. Iooss and D.D. Jospeh,Elementary Stability and Bifurcation Theory (Springer-Verlag, New York, 1980).
H.Th. Jongen, P. Jonker and F. Twilt, “On one-parameter families of sets defined by (in)equality constraints,”Nieuw Archief Voor Wiskunde, 3 (1982) 307–322.
H.Th. Jongen, P. Jonker and F. Twilt, “Critical sets in parametric optimization,”Mathematical Programming 34 (1984) 333–353.
H.Th. Jongen, P. Jonker and F. Twilt, “One-parameter families of optimization problems: Equality constraints,”Journal of Optimization Theory and Applications 48 (1986) 141–161.
H.Th. Jongen, P. Jonker and F. Twilt,Nonlinear Optimization in ℝn: 1.Morse Theory, Chebyshev Approximation (Verlag Peter Lang, New York, 1983).
T. Kato,Perturbation Theory for Linear Operators (Springer-Verlag, New York, 1966).
H.B. Keller, “Numerical solution of bifurcation and nonlinear eigenvalue problems,” in: P.H. Rabinowitz, ed.,Applications of Bifurcation Theory (Aacademic Press, New York, 1977) 359–384.
M. Kojima, “Strongly stable solutions in nonlinear programs,” in: S.M. Robinson, ed.,Analysis and Computation of Fixed Points (Academic Press, New York, 1980) 93–138.
M. Kojima and R. Hirabayashi, “Continuous deformation of nonlinear programs,” in: A.V. Fiacco, ed.,Mathematical Programming Study 21: Sensitivity, Stability and Parametric Analysis (North-Holland, Amsterdam, 1984).
T. Kupper, H.D. Mittelmann and H. Weber,Numerical Methods for Bifurcation Problems (Birkhauser-Verlag, Boston, 1984).
O.L. Mangasarian and S. Fromovitz, “The Fritz John necessary optimality conditions in the presence of equality and inequality constraints,”Journal of Mathematical Analysis and Applications 17 (1967) 37–47.
O.L. Mangasarian,Nonlinear Programming (Robert E. Krieger Publishing Company, Huntington, New York, 1979).
G.P. McCormick,Nonlinear Programming: Theory, Algorithms and Applications (John Wiley & Sons, New York, 1983).
J. Palis and F. Takens, “Stability of parametrized families of gradient vector fields,”Annals of Mathematics 118 (1983) 383–421.
W.C. Rheinboldt,Numerical Analysis of Parametrized Nonlinear Equations (John Wiley, New York, 1985).
W.C. Rheinboldt, “Solution fields of nonlinear equations and continuation methods,”SIAM Journal on Numerical Analysis 17 (1980) 221–237.
W.C. Rheinboldt and J.V. Burkardt, “A locally parametrized continuation process,”Association of Computing Machinery Transactions on Mathematical Software 9 (1983) 215–235.
S.M. Robinson, “Stability theory for systems of inequalities, part II,”SIAM Journal of Numerical Analysis 13 (1976) 497–513.
S.M. Robinson, “Strongly regular generalized equations,”Mathematics of Operations Research 5 (1980) 43–62.
Y. Sawaragi, H. Nakayama and T. Tanino,Theory of Multiobjective Optimization (Academic Press, New York, 1985).
S. Schecter, “Structure of the first-order solution set of a class of nonlinear programs with parameters,”Mathematical Programming 34 (1986) 84–110.
D. Siersma, “Singularties of functions on boundaries, corners, etc.,”Quarterly Journal of Mathematics Oxford, Second Series 32 (1981) 119–127.
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The research of this author was supported by National Science Foundation through NSF Grant
DMS-85-10201 and by the Air Force Office of Scientific Research through instrument number AFOSR-ISSA-85-00079.
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Poore, A.B., Tiahrt, C.A. Bifurcation problems in nonlinear parametric programming. Mathematical Programming 39, 189–205 (1987). https://doi.org/10.1007/BF02592952
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DOI: https://doi.org/10.1007/BF02592952