On maximum and variational principles via image space analysis
Tóm tắt
The analysis in the Image Space allows one to extend the applications of maximum and variational principles for constrained optimization. Such principles are embedded in a separation scheme, in the Image Space, which can be seen as a common root from which they are derived. In particular, Ekeland and Auchmuty Variational Principles are analysed.
Tài liệu tham khảo
Al-Homidan S., Ansari Q.H., Yao J.-C.: Some generalizations of Ekeland-type variational principle with applications to equilibrium problems and fixed point theory. Nonlinear Anal. 69(1), 126–139 (2008)
Ansari Q.H., Konnov I.V., Yao J.-C.: Existence of a solution and variational principles for vector equilibrium problems. J. Optim. Theory Appl. 110, 481–492 (2001)
Auchmuty G.: Duality for non-convex variational principles. J. Differ. Equ. 50, 80–145 (1983)
Bellman, R.: Dynamic Programming. Princeton University Press (1957)
Carathéodory, M.: Calculus of Variations and Partial Differential Equations of the First Order. Chelsea Publ. Co., New York (1982). Translation of the volume “Variationsrechnung und Partielle Differential Gleichungen Erster Ordnung”. B.G. Teubner, Berlin (1935)
Ceng L.C., Mastroeni G., Yao J.-C.: Existence of a solution and variational principles for generalized vector systems. J. Optim. Theory Appl. 137, 485–495 (2008)
Ekeland I.: Sur les problems variationels. C.R. Acad. Sci. Paris 275, 1057–1059 (1972)
Ekeland I.: On the variational principle. J. Math. Anal. Appl. 47, 324–353 (1974)
Giannessi, F.: Constrained Optimization and Image Space Analysis. Separation of Sets and Optimality Conditions, vol. 1. Springer, Berlin (2005)
Giannessi F.: Theorems of the alternative and optimality conditions. J. Optim. Theory Appl. 42, 331–365 (1984)
Giannessi F.: On the theory of Lagrangian duality. Optim. Lett. 1(1), 1–10 (2007)
Giannessi, F. (ed.): Vector Variational Inequalities and Vector Equilibria. Mathematical Theories. Kluwer, Dordrecht (2000)
Harker P.T., Pang J.S.: Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications. Math. Program. (Ser. B) 48(2), 161–220 (1990)
Hestenes M.R.: Optimization Theory: The Finite Dimensional Case. Wiley, New York (1975)
Kinderlehrer D., Stampacchia G.: An Introduction to Variational Inequalities and their Applications. Academic Press, New York (1980)
Luo H.Z., Mastroeni G., Wu X.L.: A separation approach for augmented Lagrangians in constrained nonconvex optimization. J. Optim. Theory Appl. 144, 275–290 (2010)
Rockafellar R.T., Wets R.J.-B.: Variational Analysis. Springer, Berlin (1998)
Sewell M.J.: Maximum and Minimum Principles. Cambridge University Press, Cambridge (1987)
Tardella F.: On the image of a constrained extremum problem and some applications to the existence of a minimum. J. Optim. Theory Appl. 60, 93–104 (1989)