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2014 | OriginalPaper | Buchkapitel

1. Elements of Optimization Theory and Variational Analysis

verfasst von : Alexey F. Izmailov, Mikhail V. Solodov

Erschienen in: Newton-Type Methods for Optimization and Variational Problems

Verlag: Springer International Publishing

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Abstract

In this chapter we state the problem settings that will be investigated in the book and discuss some theoretical issues necessary for the subsequent analysis of numerical methods. We emphasize that the concept of this chapter is rather minimalistic. We provide only the material that would be directly used later on and prove only those facts which cannot be regarded as “standard” (i.e., their proofs cannot be found in well-established sources). For the material that we regard as rather standard, we limit our presentation to the statements, references, and comments.

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Fußnoten
1
B is in honor of G. Bouligand.
 
Literatur
13.
Zurück zum Zitat A.V. Arutyunov, Optimality Conditions: Abnormal and Degenerate Problems (Kluwer Academic, Dordrecht, 2000)CrossRef A.V. Arutyunov, Optimality Conditions: Abnormal and Degenerate Problems (Kluwer Academic, Dordrecht, 2000)CrossRef
14.
Zurück zum Zitat A.V. Arutyunov, A.F. Izmailov, Sensitivity analysis for cone-constrained optimization problems under the relaxed constraint qualifications. Math. Oper. Res. 30, 333–353 (2005)CrossRefMATHMathSciNet A.V. Arutyunov, A.F. Izmailov, Sensitivity analysis for cone-constrained optimization problems under the relaxed constraint qualifications. Math. Oper. Res. 30, 333–353 (2005)CrossRefMATHMathSciNet
19.
Zurück zum Zitat D.P. Bertsekas, Nonlinear Programming, 2nd edn. (Athena, Belmont, 1999)MATH D.P. Bertsekas, Nonlinear Programming, 2nd edn. (Athena, Belmont, 1999)MATH
26.
Zurück zum Zitat J.F. Bonnans, Local analysis of Newton-type methods for variational inequalities and nonlinear programming. Appl. Math. Optim. 29, 161–186 (1994)CrossRefMATHMathSciNet J.F. Bonnans, Local analysis of Newton-type methods for variational inequalities and nonlinear programming. Appl. Math. Optim. 29, 161–186 (1994)CrossRefMATHMathSciNet
27.
Zurück zum Zitat J.F. Bonnans, A. Shapiro, Perturbation Analysis of Optimization Problems (Springer, New York, 2000)CrossRefMATH J.F. Bonnans, A. Shapiro, Perturbation Analysis of Optimization Problems (Springer, New York, 2000)CrossRefMATH
28.
Zurück zum Zitat J.F. Bonnans, A. Sulem, Pseudopower expansion of solutions of generalized equations and constrained optimization. Math. Program. 70, 123–148 (1995)MATHMathSciNet J.F. Bonnans, A. Sulem, Pseudopower expansion of solutions of generalized equations and constrained optimization. Math. Program. 70, 123–148 (1995)MATHMathSciNet
29.
Zurück zum Zitat J.F. Bonnans, J.Ch. Gilbert, C. Lemaréchal, C. Sagastizábal, Numerical Optimization: Theoretical and Practical Aspects, 2nd edn. (Springer, Berlin, 2006) J.F. Bonnans, J.Ch. Gilbert, C. Lemaréchal, C. Sagastizábal, Numerical Optimization: Theoretical and Practical Aspects, 2nd edn. (Springer, Berlin, 2006)
43.
44.
Zurück zum Zitat F.H. Clarke, Optimization and Nonsmooth Analysis (Wiley, New York, 1983)MATH F.H. Clarke, Optimization and Nonsmooth Analysis (Wiley, New York, 1983)MATH
46.
Zurück zum Zitat R.W. Cottle, J.-S. Pang, R.E. Stone, The Linear Complementarity Problem (SIAM, Philadelphia, 2009)CrossRefMATH R.W. Cottle, J.-S. Pang, R.E. Stone, The Linear Complementarity Problem (SIAM, Philadelphia, 2009)CrossRefMATH
62.
Zurück zum Zitat A.L. Dontchev, R.T. Rockafellar, Implicit Functions and Solution Mappings (Springer, New York, 2009)CrossRefMATH A.L. Dontchev, R.T. Rockafellar, Implicit Functions and Solution Mappings (Springer, New York, 2009)CrossRefMATH
63.
Zurück zum Zitat A.L. Dontchev, R.T. Rockafellar, Newton’s method for generalized equations: a sequential implicit function theorem. Math. Program. 123, 139–159 (2010)CrossRefMATHMathSciNet A.L. Dontchev, R.T. Rockafellar, Newton’s method for generalized equations: a sequential implicit function theorem. Math. Program. 123, 139–159 (2010)CrossRefMATHMathSciNet
65.
Zurück zum Zitat L.C. Evans, R.F. Gariepy, Measure Theory and Fine Properties of Functions (CRC Press, Boca Raton, 1992)MATH L.C. Evans, R.F. Gariepy, Measure Theory and Fine Properties of Functions (CRC Press, Boca Raton, 1992)MATH
66.
Zurück zum Zitat M. Fabian, D. Preiss, On the Clarke’s generalized Jacobian, in Proceedings of the 14th Winter School on Abstract Analysis, Circolo Matematico di Palermo, Palermo, 1987. Rendiconti del Circolo Matematico di Palermo, Ser. II, Number 14, pp. 305–307 M. Fabian, D. Preiss, On the Clarke’s generalized Jacobian, in Proceedings of the 14th Winter School on Abstract Analysis, Circolo Matematico di Palermo, Palermo, 1987. Rendiconti del Circolo Matematico di Palermo, Ser. II, Number 14, pp. 305–307
68.
Zurück zum Zitat F. Facchinei, J.-S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems (Springer, New York, 2003) F. Facchinei, J.-S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems (Springer, New York, 2003)
77.
Zurück zum Zitat D. Fernández, A.F. Izmailov, M.V. Solodov, Sharp primal superlinear convergence results for some Newtonian methods for constrained optimization. SIAM J. Optim. 20, 3312–3334 (2010)CrossRefMATHMathSciNet D. Fernández, A.F. Izmailov, M.V. Solodov, Sharp primal superlinear convergence results for some Newtonian methods for constrained optimization. SIAM J. Optim. 20, 3312–3334 (2010)CrossRefMATHMathSciNet
83.
Zurück zum Zitat A. Fischer, Solution of monotone complementarity problems with locally Lipschitzian function. Math. Program. 76, 513–532 (1997)MATH A. Fischer, Solution of monotone complementarity problems with locally Lipschitzian function. Math. Program. 76, 513–532 (1997)MATH
85.
Zurück zum Zitat A. Fischer, Local behavior of an iterative framework for generalized equations with nonisolated solutions. Math. Program. 94, 91–124 (2002)CrossRefMATHMathSciNet A. Fischer, Local behavior of an iterative framework for generalized equations with nonisolated solutions. Math. Program. 94, 91–124 (2002)CrossRefMATHMathSciNet
98.
Zurück zum Zitat J. Gauvin, A necessary and sufficient regularity conditions to have bounded multipliers in nonconvex programming. Math. Program. 12, 136–138 (1977)CrossRefMATHMathSciNet J. Gauvin, A necessary and sufficient regularity conditions to have bounded multipliers in nonconvex programming. Math. Program. 12, 136–138 (1977)CrossRefMATHMathSciNet
113.
120.
Zurück zum Zitat J.-B. Hiriart-Urruty, J.-J. Strodiot, V.H. Nguyen, Generalized Hessian matrix and second-order optimality conditions for problems with C 1, 1 data. Appl. Math. Optim. 11, 43–56 (1984)CrossRefMATHMathSciNet J.-B. Hiriart-Urruty, J.-J. Strodiot, V.H. Nguyen, Generalized Hessian matrix and second-order optimality conditions for problems with C 1, 1 data. Appl. Math. Optim. 11, 43–56 (1984)CrossRefMATHMathSciNet
128.
Zurück zum Zitat A.F. Izmailov, On the analytical and numerical stability of critical Lagrange multipliers. Comput. Math. Math. Phys. 45, 930–946 (2005)MathSciNet A.F. Izmailov, On the analytical and numerical stability of critical Lagrange multipliers. Comput. Math. Math. Phys. 45, 930–946 (2005)MathSciNet
129.
Zurück zum Zitat A.F. Izmailov, Sensitivity of solutions to systems of optimality conditions under the violation of constraint qualifications. Comput. Math. Math. Phys. 47, 533–554 (2007)CrossRefMathSciNet A.F. Izmailov, Sensitivity of solutions to systems of optimality conditions under the violation of constraint qualifications. Comput. Math. Math. Phys. 47, 533–554 (2007)CrossRefMathSciNet
130.
Zurück zum Zitat A.F. Izmailov, Solution sensitivity for Karush–Kuhn–Tucker systems with nonunique Lagrange multipliers. Optimization 59, 747–775 (2010)CrossRefMATHMathSciNet A.F. Izmailov, Solution sensitivity for Karush–Kuhn–Tucker systems with nonunique Lagrange multipliers. Optimization 59, 747–775 (2010)CrossRefMATHMathSciNet
142.
Zurück zum Zitat A.F. Izmailov, M.V. Solodov, A note on solution sensitivity for Karush–Kuhn–Tucker systems. Math. Meth. Oper. Res. 61, 347–363 (2005)CrossRefMATHMathSciNet A.F. Izmailov, M.V. Solodov, A note on solution sensitivity for Karush–Kuhn–Tucker systems. Math. Meth. Oper. Res. 61, 347–363 (2005)CrossRefMATHMathSciNet
155.
Zurück zum Zitat A.F. Izmailov, A.S. Kurennoy, M.V. Solodov, A note on upper Lipschitz stability, error bounds, and critical multipliers for Lipschitz-continuous KKT systems. Math. Program. 142, 591–604 (2013)CrossRefMATHMathSciNet A.F. Izmailov, A.S. Kurennoy, M.V. Solodov, A note on upper Lipschitz stability, error bounds, and critical multipliers for Lipschitz-continuous KKT systems. Math. Program. 142, 591–604 (2013)CrossRefMATHMathSciNet
166.
Zurück zum Zitat W. Karush, Minima of functions of several variables with inequalities as side constraints. Technical Report (master’s thesis), Department of Mathematics, University of Chicago, 1939 W. Karush, Minima of functions of several variables with inequalities as side constraints. Technical Report (master’s thesis), Department of Mathematics, University of Chicago, 1939
168.
169.
Zurück zum Zitat D. Klatte, B. Kummer, Nonsmooth Equations in Optimization: Regularity, Calculus, Methods and Applications (Kluwer Academic, Dordrecht, 2002) D. Klatte, B. Kummer, Nonsmooth Equations in Optimization: Regularity, Calculus, Methods and Applications (Kluwer Academic, Dordrecht, 2002)
170.
Zurück zum Zitat D. Klatte, K. Tammer, On the second order sufficient conditions to perturbed C 1, 1 optimization problems. Optimization 19, 169–180 (1988)CrossRefMATHMathSciNet D. Klatte, K. Tammer, On the second order sufficient conditions to perturbed C 1, 1 optimization problems. Optimization 19, 169–180 (1988)CrossRefMATHMathSciNet
171.
Zurück zum Zitat M. Kojima, Strongly stable stationary solutions in nonlinear programs, in Analysis and Computation of Fixed Points, ed. by S.M. Robinson (Academic, NewYork, 1980), pp. 93–138 M. Kojima, Strongly stable stationary solutions in nonlinear programs, in Analysis and Computation of Fixed Points, ed. by S.M. Robinson (Academic, NewYork, 1980), pp. 93–138
174.
Zurück zum Zitat H.W. Kuhn, A.W. Tucker, Non-linear programming. in Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, ed. by J. Neyman (University of California Press, Berkeley, 1951), pp. 481–493 H.W. Kuhn, A.W. Tucker, Non-linear programming. in Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, ed. by J. Neyman (University of California Press, Berkeley, 1951), pp. 481–493
187.
Zurück zum Zitat O.L. Mangasarian, S. Fromovitz, The Fritz John necessary optimality conditions in the presence of equality and inequality constraints. J. Math. Anal. Appl. 17, 37–47 (1967)CrossRefMATHMathSciNet O.L. Mangasarian, S. Fromovitz, The Fritz John necessary optimality conditions in the presence of equality and inequality constraints. J. Math. Anal. Appl. 17, 37–47 (1967)CrossRefMATHMathSciNet
199.
201.
Zurück zum Zitat B.S. Mordukhovich, Variational Analysis and Generalized Differentiation (Springer, Berlin, 2006) B.S. Mordukhovich, Variational Analysis and Generalized Differentiation (Springer, Berlin, 2006)
208.
Zurück zum Zitat J. Nocedal, S.J. Wright, Numerical Optimization, 2nd edn. (Springer, New York, 2006)MATH J. Nocedal, S.J. Wright, Numerical Optimization, 2nd edn. (Springer, New York, 2006)MATH
222.
224.
Zurück zum Zitat L. Qi, H. Jiang, Semismooth Karush–Kuhn–Tucker equations and convergence analysis of Newton and quasi-Newton methods for solving these equations. Math. Oper. Res. 22, 301–325 (1997)CrossRefMATHMathSciNet L. Qi, H. Jiang, Semismooth Karush–Kuhn–Tucker equations and convergence analysis of Newton and quasi-Newton methods for solving these equations. Math. Oper. Res. 22, 301–325 (1997)CrossRefMATHMathSciNet
227.
233.
Zurück zum Zitat S.M. Robinson, Stability theory for systems of inequalities, Part II. Differentiable nonlinear systems, SIAM J. Numer. Anal. 13, 497–513 (1976) S.M. Robinson, Stability theory for systems of inequalities, Part II. Differentiable nonlinear systems, SIAM J. Numer. Anal. 13, 497–513 (1976)
239.
250.
Zurück zum Zitat M.V. Solodov, Constraint qualifications, in Wiley Encyclopedia of Operations Research and Management Science, ed. by J.J. Cochran (Wiley, London, 2010) M.V. Solodov, Constraint qualifications, in Wiley Encyclopedia of Operations Research and Management Science, ed. by J.J. Cochran (Wiley, London, 2010)
Metadaten
Titel
Elements of Optimization Theory and Variational Analysis
verfasst von
Alexey F. Izmailov
Mikhail V. Solodov
Copyright-Jahr
2014
DOI
https://doi.org/10.1007/978-3-319-04247-3_1

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