High-order approximations and generalized necessary conditions for optimality

SIAM Journal on Control and Optimization

Volumn 37, Issue 1, 1999, Pages 33-53

  Ledzewicz, Urszula ^a   Schättler, Heinz ^b  

^a SOUTHERN ILLINOIS UNIVERSITY EDWARDSVILLE   (United States)

^b WASHINGTON UNIVERSITY   (United States)

Author keywords

Abnormal processes; High order necessary conditions; High order necessary conditions for optimality; High order tangent sets; Lyusternik theorem

Indexed keywords

APPROXIMATION THEORY; CONSTRAINT THEORY; EQUATIONS OF MOTION; MATHEMATICAL OPERATORS; PROBLEM SOLVING;

EQUALITY CONSTRAINTS; HIGH-ORDER APPROXIMATIONS; INEQUALITY CONSTRAINTS; LYUSTERNIK THEOREM;

OPTIMAL CONTROL SYSTEMS;

EID: 0032154102     PISSN: 03630129     EISSN: None     Source Type: Journal    
DOI: 10.1137/s0363012997317748     Document Type: Article

References (25)

1. A. V. ARUTYUNOV, Higher-order conditions in abnormal extremal problems with constraints of equality type, Soviet Math. Dokl., 42 (1991), pp. 799-804.
2. A. V. ARUTYUNOV, Second-order conditions in extremal problems with a finite-dimensional image. 2-normal mappings. Izv. Ross. Akad. Nauk Ser. Mat., 60 (1996), pp. 37-62 (in Russian).
3. E. R. AVAKOV, Extremum conditions for smooth problems with equality-type constraints, USSR Comput. Math. and Math. Phys., 25 (1985), pp. 24-32 (translated from Zh. Vychisl. Mat. i Fiz., 25 (1985)).
4. E. R. AVAKOV, Extremum conditions for smooth problems with equality-type constraints, USSR Comput. Math. and Math. Phys., 25 (1985), pp. 24-32 (translated from Zh. Vychisl. Mat. i Fiz., 25 (1985)).
5. E. R. AVAKOV, Necessary conditions for a minimum for nonregular problems in Banach spaces. Maximum principle for abnormal problems in optimal control, Trudy Mat. Inst. Akad. Nauk. SSSR, 185 (1988), pp. 3-29 (in Russian).
6. E. R. AVAKOV, Necessary extremum conditions for smooth abnormal problems with equality-and inequality constraints, Mat. Zametki, 45 (1989), pp. 3-11 (in Russian). Translated in J. Math. Notes, 45 (1989), pp. 431-437.
7. E. R. AVAKOV, Necessary extremum conditions for smooth abnormal problems with equality- and inequality constraints, Mat. Zametki, 45 (1989), pp. 3-11 (in Russian). Translated in J. Math. Notes, 45 (1989), pp. 431-437.
8. A. BEN-TAL AND J. ZOWE, A unified theory of first and second order conditions for extremum problems in topological vector spaces, Mathematical Programming Study, 19 (1982), pp. 39-76.
9. A. V. DMITRUK, Quadratic conditions for a Pontryagin minimum in an optimal control problem linear in the control. II. Theorems on weakening equality constraints, Math. USSR Izvestiya, 31 (1986), pp. 121-141.
10. I. V. GIRSANOV, Lectures on Mathematical Theory of Extremum Problems, Lecture Notes in Econom. and Math. Systems 67, Springer-Verlag, Berlin, 1972.
11. K. H. HOFFMANN AND H. J. KORNSTAEDT, Higher-order necessary conditions in abstract mathematical programming, J. Optim. Theory Appl., 26 (1978), pp. 533-568.
12. A. D. IOFFE, Necessary and sufficient conditions for a local minimum. 3: Second-order conditions and augmented duality, SIAM J. Control Optim., 17 (1979), pp. 266-288.
13. A. D. IOFFE AND V. M. TIKHOMIROV, Theory of Extremal Problems, North-Holland, Amsterdam, The Netherlands, 1979.
14. A. F. IZMAILOV, Optimality conditions for degenerate extremum problems with inequality-type constraints, Comput. Math. Math. Phys., 34 (1994), pp. 723-736.
15. U. LEDZEWICZ, An extension of the local maximum principle to abnormal optimal control problems, J. Optim. Theory Appl., 77 (1993), pp. 661-681.
16. U. LEDZEWICZ AND H. SCHÄTTLER, Second-order conditions for extremum problems with non-regular equality constraints, J. Optim. Theory Appl., 86 (1995), pp. 113-144.
17. U. LEDZEWICZ AND H. SCHÄTTLER, An extended maximum principle, Nonlinear Anal., 29 (1997), pp. 159-183.
18. U. LEDZEWICZ AND H. SCHÄTTLER, A high-order generalization of the Lyusternik theorem, Nonlinear Anal., 34 (1998), pp. 793-815.
19. U. LEDZEWICZ AND H. SCHÄTTLER, A high-order generalization of the Lyusternik theorem and its application to optimal control problems, Dynamical Systems and Differential Equations, II, W. Chen and S. Hu, eds., Missouri State University, 1998, pp. 45-59.
20. U. LEDZEWICZ AND H. SCHÄTTLER, High-order extended maximum principles for optimal control problems with non-regular constraints, in Optimal Control: Theory, Algorithms and Applications, W. W. Hager and P. M. Pardalos, eds., Kluwer Academic Publishers, Dordrecht, The Netherlands, 1998, pp. 298-325.
21. E. S. LEVITIN, A. A. MILYUTIN, AND N. P. OSMOLOVSKII, Conditions of higher order for a local minimum in problems with constraints, Russian Math. Surveys, 33 (1978), pp. 97-168.
22. A. A. MILYUTIN, On quadratic conditions for an extremum in smooth problems with a finite-dimensional image, in Methods of the Theory of Extremal Problems in Economics, V. L. Levin, ed., Nauka, Moscow, 1981, pp. 138-177 (in Russian).
23. ZS. PÁLES AND V. ZEIDAN, Nonsmooth optimum problems with constraints, SIAM J. Control Optim., 32 (1994), pp. 1476-1502.
24. ZS. PÁLES AND V. ZEIDAN, First- and second-order necessary conditions for control problems with constraints, Trans. Amer. Math. Soc., 346 (1994), pp. 421-453.
25. A. A. TRETYAKOV, Necessary and sufficient conditions for optimality of p-th order, USSR Comput. Math. Math. Phys., 24 (1984), pp. 123-127.