A Characterization of Maximal Monotone Operators
Tóm tắt
It is shown that a set-valued map
$M:\mathbb{R}^{q} \rightrightarrows \mathbb{R}^{q}$
is maximal monotone if and only if the following five conditions are satisfied: (i) M is monotone; (ii) M has a nearly convex domain; (iii) M is convex-valued; (iv) the recession cone of the values M(x) equals the normal cone to the closure of the domain of M at x; (v) M has a closed graph. We also show that the conditions (iii) and (v) can be replaced by Cesari’s property (Q).
Tài liệu tham khảo
Borwein, J., Lewis, A.: Convex Analysis and Nonlinear Optimization. Springer, New York (2000)
Cesari, L.: I: existence theorems for weak and usual optimal solutions in Lagrange problems with unilateral constraints. II: existence theorems for weak solutions. Trans. Amer. Math. Soc. 124, 369–412, 413–430 (1966)
Cesari, L.: Optimization—Theory and Applications. Problems with Ordinary Differential Equations, Applications of Mathematics, vol. 17. Springer, Berlin Heidelberg New York (1983)
Cesari, L., Suryanarayana, M.B.: Convexity and property (Q) in optimal control theory. SIAM J. Control 12, 705–720 (1974)
Hou, S.H.: On property (Q) and other semi-continuity properties of multifunctions. Pacific J. Math. 103(1), 39–56 (1982)
Löhne, A., Zălinescu, C.: On convergence of closed convex sets. J. Math. Anal. Appl. 319(2), 617–634 (2006)
Löhne, A.: On semicontinuity of convex-valued multifunctions and Cesari’s property (Q). J. Convex Anal. (2007) (submitted)
Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)
Suryanarayana, M.B.: Upper semicontinuity of set-valued functions. J. Optim. Theory Appl. 41(1), 185–211(1983)