Abstract
A survey of stability and sensitivity results for the solutions to parameter depenedent cone constrained optimization problems in abstract Banach spaces is presented. An application to optimal control problems for nonlinear ordinary differential equations subject to control and state constraints is given.
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References
Allgower E.L., Georg K., „Numerical Continuation Methods. An Introduction,“ Springer Series in Computational Mathematics Vol 13, Springer Verlag, Berlin, 1990.
Alt W., “On the approximation of infinite optimization problems with an application to optimal control problems,” Appl. Math. Optim. 12 (1984), 15–27.
Alt W., “Stability of solutions for a class of nonlinear cone constrained optimization problems, Part 1: Basic theory,” Numer. Funct. Anal. and Optimiz. 10 (1989), 1053–1064.
Alt W., “Stability of solutions for a class of nonlinear cone constrained optimization problems, Part 2: Application to parameter estimation,” Numer. Funct. Anal. and Optimiz. 10 (1989), 1065–1076.
Alt W., “Stability for parameter estimation in two point boundary value problems,” Optimization 22 (1991), 99–111.
Alt W., “The Lagrange-Newton method for infinite-dimensional optimization problems,” Numer. Funct. Anal. and Optimiz. 11 (1990), 201–224.
Alt W., “Parametric programming with applications to optimal control and sequential quadratic programming,” Bayreuther Mathematische Schriften 35 (1990), 1–37.
Alt W., „Stability of Solutions and the Lagrange-Newton Method for Nonlinear Optimization and Optimal Control Problems,” (Habilitationsschrift), Universität Bayreuth, Bayreuth, 1990.
Alt W., Malanowski K., „The Lagrange-Newton method for nonlinear optimal control problems”, Comput. Optim. and Applications 2 (1993), 77–100.
Alt W., Mackenroth U., „Convergence of finite element approximations to state constrained comvex boundary control problems,” SIAM J. Control and Optimization 27 (1987), 718–736.
Alt W., Sontag R. Tröltzsch F., „An SQP method for optimal control of weakly singular Hammerstein integral equations”, Deutsche Forschungsgemeinschaft, SPP „Anwendungsbezogene Optimierung und Steuerung”, Report No. 423, Chemnitz, 1993.
Bank B., Guddat J., Klatte D., Kummer B., Tammer K., „Non-linear Parametric Optimization” Birkhäser Verlag, Basel, 1983.
Barbet L., „Lipschitzian properties of the optimal solutionns for parametric variational inequalities in Banach spaces” (to be published)
Bock H. G., „Zur numerischen Behandlung zuschtandsbeschränkter Steuerungsprobleme mit Mehrzielmethode und Homotopierverfahren, ZAMM 57 (1977), T 266–T 268.
Bock H. G., „Randwertproblemmethoden zur Parameteridentifizierung in Systemen nichtlinearer Differentialgleichungen,” Bonner Mathematische Schriften 183, Bonn, 1987.
Bonnans J.-F., Sulem A., „Pseudopower expansion of solutions of generalized equations and constrained optimization problems” (to be published).
Brdyś M., „Theory of Hierarchical Contronl Systems for Complex Slowlyvarying Processes,” Politechnika Warszawska, Prace Naukowe, Elektronika, z 47, Warszawa, 1980 (in Polish).
Chao G. S., Friesz T. L., „Spacial price equilibrium sensitivity analysis,” Transportation Research 18B (1984), 423–440.
Colonius F., Kunisch K., “Stability for parameter estimation in two point boundary value problems,” Journal für die Reine und Angewandte Mathematik 370 (1986), 1–29.
Colonius F., Kunisch K., “Stability of perturbed optimization problems with applications to parameter estimation,” Report Nr. 205, Institut für Dynamische Systeme, Universität Bremen, 1989.
Defermos S., Nagurney A., „Sensitivity analysis for the asymetric network equilibrium problem,” Math. Programming 28 (1984), 174–184.
Dontchev A. L., “Perturbations, Approximations and Sensitivity Analysis of Optimal Control Systems,” Lecture Notes in Control and Information Sciences Vol. 52, Springer Verlag, Berlin-Heidelberg-New York, 1983.
Dontchev A. L., Hager W. W., „Lipschitz stability in nonlinear control and optimization”, SIAM J. Control and Optimization 31 (1993), 569–603.
Dontchev A. L., Hager W. W., Poore A B., Yang B., „Optimality, stability and convergence in nonlinear control” (to be published).
Fiacco A. V., “Sensitivity analysis for nonlinear programming using penalty methods,” Mathematical Programming 10 (1976), 287–311.
Fiacco A. V., “Introduction to sensitivity and stability analysis in nonlinear programming,” Academic Press, New York-London, 1983.
Findeisen W., Bailey F. N., Brdyś M., Malinowski K., Tatjewski P., Woźniak A., „Control and Coordination in Hierarchical System,” J. Wiley & Sons, Chichester-New York-Brisbane-Toronto, 1980.
Friesz T. L., Tobin R. L., Miller T., „Existence theory for spacially competitive network facility location models.” Annals Oper. Res. 18 (1989), 267–276.
Gahutu W. H., Looze D. P., „Parametric coordination in hierarchical control,” Large Scale Systems 8 (1985), 33–45.
Gfrerer H., Guddat J., Wacker HJ., „A globally convergent algorithm based on imbedding and parametric optimization,” Computing 30 (1983), 225–252.
Guddat J., Guerra-Vazques F., Jongen H. Th., „Singularities, Path Following and Jumps,” J. Wiley & Sons, Chichester, 1990.
Hager W. W., “Lipschitz continuity for constrained processes,” SIAM J. Control and Optimization 17 (1979), 321–337.
Hager W. W., “Approximations to the multiplier methods,” SIAM J. Numer. Anal. 22 (1985), 16–46.
Hager W. W., “Multiplier methods for nonlinear optimal control,” SIAM J. Numer. Anal. 27 (1990), 1061–1080.
Hager W. W., Ianculescu, G. D., “Dual approximations in optimal control,” SIAM J. Control and Optimization 22 (1984), 423–465.
Haraux A., „How to differentiate the projection on a convex set in Hilbert space. Some applications to variational inequalities,” J.Math.Soc. Japan 29 (1977), 615–632.
Haslinger J., Neittaanmäki P., „Finite Element Approximation for Optimal Shape Design: Theory and Applications,” J. Wiley $ Sons, Chichester, 1988.
Ioffe A., „On sensitivity analysis of nonlinear programs in Banach spaces: the approach via composite unconstrained optimization,” to appear in SIAM J. Optimization.
Ito K., Kunisch K., “Sensitivity analysis of solutions to optimization problems in Hilbert spaces with applications to optimal control and estimation,” J. Diff. Equations 99, 1–40.
Ito K., Kunisch K., “The augmented Lagrangian method for equality and inequality constraints in Hilbert spaces,” Mathematical Programming 46 (1990), 341–360.
Ito K., Kunisch K., “The augmented Lagrangian method for parameter estimation in elliptic systems,” SIAM J. Control and Optimization 28 (1990), 113–136.
Ito K., Kunisch K., “An augmented Lagrangian technique for variational inequalities,” Appl. Math. Optim. 21 (1990), 223–241.
de Jong J. L., Machielsen K. C. P., “On the application of sequential quadratic programming to state constrained optimal control,” in: IFAC, Control Applications of Nonlinear Programming, Capri, Italy, 1985.
Kall P., „On approximations and stability in stochastic programming,” in: J. Guddat, H. Th. Jongen, B. Kummer B., F. Nozicka, (eds), Parametric Optimization and Related Topics, Academie Verlag, Berlin, 1987.
Kelley C. T., Wright S. J., „Sequential quadratic programming for certain parameter identification problems” (to be published).
Kruskal J. B., „Two convex counterexamples: a discontinuous envelope function and a nondifferentiable nearest-point mapping,” Proc. Amer. Math. Soc. 23 (1969), 697–703.
Lasiecka I., „Boudary control of parabolic systems: finite element approximation,” Appl. Math. Optim. 6 (1980), 31–62.
Lasiecka I., „Ritz-Galerkin approximation of the time optimal boudary control problem for parabolic system with Dirichlet boundary conditions,” SIAM J. Control and Optimization 22 (1984), 477–500.
Levitin E.S., “On the local perturbation theory of a problem of mathematical programming in a Banach space,” Soviet Math. Dokl. 16 (1975), 1354–1358.
Levitin E. S., “Differentiability with respect to a parameter of the optimal value in parametric problems of mathematical programming,” Cybernetics 12 (1976), 46–64.
Levitin E. S., „Perturbation Theory in Mathematical Programming and its Applications,” Nauka, Moscow, 1992, (in Russian), to be published in English by J. Wiley & Sons.
Lundberg B. N., Poore A. B., „Numerical continuation and singularity detection methods for parametric nonlinear programming,” SIAM J. Optim. 3 (1993), 134–154.
Machielsen K, C. P., “Numerical solution of optimal control problems with state constraints by sequential quadratic programming in function space,” CWI Tract 53, Amsterdam, 1987.
Malanowski K., „Convergence of approximations vs. regularity of solutions for convex, control-constrained optimal control problems,” Appl. Math. Opt. 8 (1981), 69–95.
Malanowski K., “Second order conditions and constraint qualifications in stability and sensitivity analysis of solutions to optimization problems in Hilbert spaces,” Appl. Math. Opt. 25 (1992) 51–79.
Malanowski K., „Two-norm approach in stability and sensitivity analysis of optimization and optimal control problems,” to appear in Advances in Math. Sc. Appl.
Malanowski K., „Stability and sensitivity of solutions to nonlinear optimal control problems”, to appear in Appl. Math. Optim.
Malanowski K., „Regularity of solutions in stability analysis of optimization and optimal control problems.”, to appear in Control and Cybernetics.
Marcotte P., „Network design with congestion effects: a case of bilevel programming,” Math. Programming 34 (1986), 142–162.
Maurer H., “First and second order sufficient optimality conditions in mathematical programming and optimal control,” Mathematical Programming Study 14 (1981), 163–177.
Maurer H., Pesch H. J., „Solution differentiability for nonlinear parametric control problems,” Deutsche Forschungsgemeinschaft, SPP „Anwendungsbezogene Optimierung und Steuerung,” Report No. 316, München, 1991.
Maurer H., Zowe J., „First-and second order sufficient optimality conditions for infinite-dimensional programming problems,” Math. Programming 16 (1979), 98–110.
Mignot F., „Contrôle dans les inéquations variationelles elliptiques,” J. Funct. Anal. 22 (1976), 130–185.
Qutrata J. V., „On the numerical solution of a class of Stackelberg problems,” Zeit. Oper. Res. 4 (1990), 255–278.
Outrata J. V., „On necessary optimality conditions for Stackelberg problems,” J. Optim. Theory Appl. 76 (1993), 305–320.
Robinson S. M., “Perturbed Kuhn-Tucker points and rates of convergence for a class of nonlinear-programming algorithms,” Mathematical Programming 7 (1974), 1–16.
Robinson S. M., „Stability theory for system of inequalities, Part II: Differentiable nonlinear systems, SIAM J. Numer. Anal. 13 (1976), 497–513.
Robinson S. M., “Strongly regular generalized equations,” Mathematics of Operatios Research 5 (1980), 43–62.
Robinson S. M., Wets R. J.-B., „Stability in two-stage stochasic programming,” SIAM J. Control and Optimization 25 (1987), 1409–1416.
Römisch W., Schultz R., „Stability analysis for stochastic programs,” Annals of Operations Research 30 (1991), 241–266.
Römisch W., Schultz R., „Lipschitz stability for stochastic programs with complete recourse,” Deutsche Forschungsgemeinschaft, SPP „Anwendungsbezogene Optimierung und Steuerung,” Report No. 408, Berlin, 1992d. (to be published).
Shapiro A., „Asymptotic properties of statistical estimators in stochastic programming,” The Annals of Statistics 17 (1989), 841–858.
Shapiro A., „On differential stability in stochastic programming,” Math. Programming 47 (1991), 107–186.
Shapiro A., „Asymptotic analysis of stochastic programs,” Annals of Oper. Res. 30 (1991), 169–186.
Shapiro A., „Perturbation analysis of optimization in Banach spaces,” Numerical Funct. Anal. Optim. 13 (1992), 97–116.
Shapiro A., „Asymptotic behavior of optimal solutions in stochastic programming,” to appear in Math. Oper. Res..
Shapiro A., „Sensitivity analysis of parametrized programs via generalized equations,” (to be published).
Shapiro A., Bonnans J. F., „Sensitivity analysis of parametrized programs under cone constraints,” SIAM J. Control and Optimization 30 (1992), 1409–1421.
Shimizu K., Ishizuka Y., „Optimality conditions and algorithms for parameter design problems with two-level structure,” IEEE Trans. Aut. Contr. AC-30 (1985), 986–993.
Sokolowski J., Zolesio J.-P., „Sensitivity Analysis in Shape Optimization,” Springer Series in Computational Mathematics, Vol 16, Springer-Verlag, Berlin, 1992.
Tobin R. L., Friesz T. L., „Spacial competition facility location models: definition, formulation and solution approach,” Annals Oper. Res. 6 (1986), 49–74.
Tröltzsch F., “Semdiscrete finite element approximation of parabolic boundary control problems-convergence of switching points,” in: Optimal Control of Partial Differential Equations II, Int. Ser. Num. Math., Vol. 78, Birkhäser, Basel, 1987.
Tröltzsch F., „Approximation of nonlinear parabolic boundary control problems by the Fourier method — convergence of optimal controls,” Optimization 22 (1991), 83–98.
Tröltzsch F., „Semidiscrete Ritz-galerkin approximation of nonlinear parabolic boundary control problems-strong convergence of optimal controls,” Deutsche Forschungsgemeinschaft, SPP „Anwendungsbezogene Optimierung und Steuerung,” Report No. 325, Augsburg, 1991.
Tröltzsch F., „On convergence of semidiscrete Ritz-Galerkin schemes applied to boundary control problems of parabolic equations with non-linear boundary conditions,” ZAMM 72 (1992), 291–301.
Ursescu C., „Multifunctions with closed convex graph,” Czechoslovak Math. J. 25 (1975), 438–441.
Yang B., „Some Numerical Methods for a Class of Nonlinear Optimal Control Problems,” (dissertation) Colorado State University, Fort Collins, Colorado, 1991.
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Malanowski, K. (1994). Stability and sensitivity analysis of solutions to infinite-dimensional optimization problems. In: Henry, J., Yvon, JP. (eds) System Modelling and Optimization. Lecture Notes in Control and Information Sciences, vol 197. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0035462
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DOI: https://doi.org/10.1007/BFb0035462
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