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1.
Some minimax problems of vector-valued functions   总被引:2,自引:0,他引:2  
The concepts of cone extreme points, cone saddle points, and cone saddle values are introduced. The relation of inclusion among the sets mini xX max yY f(x, y), maxi yY min xX f(x, y), and the set of all weak cone saddle values is investigated in the case where the image space n off is ordered by an acute convex cone.The author is grateful for the useful suggestions and comments given by Prof. K. Tanaka, Niigata University, Niigata, Japan.The author would like to thank the referees for their valuable suggestions on the original draft.  相似文献   

2.
Given a continuous mapF:R n R n and a lower semicontinuous positively homogeneous convex functionh:R n R, the nonlinear complementarity problem considered here is to findxR + n andyh(x), the subdifferential ofh atx, such thatF(x)+y0 andx T (F(x)+y)=0. Some existence theorems for the above problem are given under certain conditions on the mapF. An application to quasidifferentiable convex programming is also shown.The authors are grateful to Professor O. L. Mangasarian and the referee for their substantive suggestions.  相似文献   

3.
In this paper, we deal with the following stability problem: given a differential inclusion of the formx'F(t,x,), where is a parameter varying in a topological space , find conditions under which the set of all , such that the differential inclusion is controllable, is open in . Applying Theorem 3.1 of Ref. 3, we get a result in this direction, assuming, as leader hypotheses, thatF(t,·,) is a convex process, from into itself, and thatF(t,x,·) is lower semicontinuous.  相似文献   

4.
The projected gradient methods treated here generate iterates by the rulex k+1=P (x k s k F(x k )),x 1 , where is a closed convex set in a real Hilbert spaceX,s k is a positive real number determined by a Goldstein-Bertsekas condition,P projectsX into ,F is a differentiable function whose minimum is sought in , and F is locally Lipschitz continuous. Asymptotic stability and convergence rate theorems are proved for singular local minimizers in the interior of , or more generally, in some open facet in . The stability theorem requires that: (i) is a proper local minimizer andF grows uniformly in near ; (ii) –F() lies in the relative interior of the coneK of outer normals to at ; and (iii) is an isolated critical point and the defect P (xF(x)) –x grows uniformly within the facet containing . The convergence rate theorem imposes (i) and (ii), and also requires that: (iv)F isC 4 near and grows no slower than x4 within the facet; and (v) the projected Hessian operatorP F 2 F()F is positive definite on its range in the subspaceF orthogonal toK . Under these conditions, {x k } converges to from nearby starting pointsx 1, withF(x k ) –F() =O(k –2) and x k – =O(k –1/2). No explicit or implied local pseudoconvexity or level set compactness demands are imposed onF in this analysis. Furthermore, condition (v) and the uniform growth stipulations in (i) and (iii) are redundant in n .  相似文献   

5.
Generalized convex functions and vector variational inequalities   总被引:3,自引:0,他引:3  
In this paper, (, ,Q)-invexity is introduced, where :X ×X intR m + , :X ×X X,X is a Banach space,Q is a convex cone ofR m . This unifies the properties of many classes of functions, such asQ-convexity, pseudo-linearity, representation condition, null space condition, andV-invexity. A generalized vector variational inequality is considered, and its equivalence with a multi-objective programming problem is discussed using (, ,Q)-invexity. An existence theorem for the solution of a generalized vector variational inequality is proved. Some applications of (, ,Q)-invexity to multi-objective programming problems and to a special kind of generalized vector variational inequality are given.The author is indebted to Dr. V. Jeyakumar for his constant encouragement and useful discussion and to Professor P. L. Yu for encouragement and valuable comments about this paper.  相似文献   

6.
This paper considers analogues of the Helmholtz projections of the set of selections of a piecewise smooth multivalued map , n2. It is shown that, for mn–1 (m=1), the closure of the projection of on the subspace of gradient fields (solenoidal vector fields) is a convex set. For the general case, there are given point-wise conditions on the values of the map which ensure that the closure of the projection of contains the zero element. Possible applications to optimal control problems are discussed.  相似文献   

7.
Summary We show, among other things, that the positive zeros of a solution ofy +x y=0,y(0)=0 decrease to 1 as increases, 0.
Sommario Si dimostra, tra l'altro, che gli zeri positivi d'una soiuzione diy +x y=0,y(0)=0 decrescono al limite 1, quando cresce, 0.


To the memory of Milo Háik

This research was supported by grants from the Natural Sciences and Engineering Research Council (Canada) and Consiglio Nazionale delle Ricerche (Italy). Some of the work was done while the second-named author was visiting the Department of Mathematics, University of Torino.  相似文献   

8.
Let be a probability measure on n 2 × 2 stochastic matrices, n an arbitrary positive integer, and = (w) lim n n , such that the support of consists of 2 × 2 stochastic matrices of rank one, and as such, can be regarded as a probability measure on [0, 1]. We present simple sufficient conditions for to be continuous singular w.r.t. the Lebesgue measure on [0, 1]. We also determine , given .  相似文献   

9.
We prove that if aC 1 smooth change of variable : generates a bounded composition operatorff° in the spaceA p()=L p ,p2, then is linear (affine).We also prove that for a nonlinearC 1 mapping , the norms of exponentialse i as Fourier multipliers inL p () tend to infinity (,||). In both results the condition C 1 is sharp, it cannot be replaced by the Lipschitz condition.  相似文献   

10.
In this paper we consider the problem of determining and constructing E- and MV-optimal block designs to use in experimental settings where treatments are applied to experimental units occurring in b blocks of size k, k. It is shown that some of the well-known methods for constructing E- and MV-optimal unequally replicated designs having k fail to yield optimal designs in the case where . Some sufficient conditions are derived for the E- and MV-optimality of block designs having and methods for constructing designs satisfying these sufficient conditions are given.  相似文献   

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