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1.
《Optimization》2012,61(1):39-50
We extend the convergence analysis of a smoothing method [M. Fukushima and J.-S. Pang (2000). Convergence of a smoothing continuation method for mathematical programs with complementarity constraints. In: M. Théra and R. Tichatschke (Eds.), Ill-posed Variational Problems and Regularization Techniques, pp. 99–110. Springer, Berlin/Heidelberg.] to a general class of smoothing functions and show that a weak second-order necessary optimality condition holds at the limit point of a sequence of stationary points found by the smoothing method. We also show that convergence and stability results in [S. Scholtes (2001). Convergence properties of a regularization scheme for mathematical programs with complementarity constraints. SIAM J. Optim., 11, 918–936.] hold for a relaxation problem suggested by Scholtes [S. Scholtes (2003). Private communications.] using a class of smoothing functions. In addition, the relationship between two technical, yet critical, concepts in [M. Fukushima and J.-S. Pang (2000). Convergence of a smoothing continuation method for mathematical programs with complementarity constraints. In: M. Théra and R. Tichatschke (Eds.), Ill-posed Variational Problems and Regularization Techniques, pp. 99–110. Springer, Berlin/Heidelberg; S. Scholtes (2001). Convergence properties of a regularization scheme for mathematical programs with complementarity constraints. SIAM J. Optim., 11, 918–936.] for the convergence analysis of the smoothing and regularization methods is discussed and a counter-example is provided to show that the stability result in [S. Scholtes (2001). Convergence properties of a regularization scheme for mathematical programs with complementarity constraints. SIAM J. Optim., 11, 918–936.] cannot be extended to a weaker regularization.  相似文献   

2.
Summary. Some recent investigations (see e.g., Gerstner and Griebel [5], Novak and Ritter [9] and [10], Novak, Ritter and Steinbauer [11], Wasilkowski and Woźniakowski [18] or Petras [13]) show that the so-called Smolyak algorithm applied to a cubature problem on the d-dimensional cube seems to be particularly useful for smooth integrands. The problem is still that the numbers of nodes grow (polynomially but) fast for increasing dimensions. We therefore investigate how to obtain Smolyak cubature formulae with a given degree of polynomial exactness and the asymptotically minimal number of nodes for increasing dimension d and obtain their characterization for a subset of Smolyak formulae. Error bounds and numerical examples show their good behaviour for smooth integrands. A modification can be applied successfully to problems of mathematical finance as indicated by a further numerical example. Received September 24, 2001 / Revised version received January 24, 2002 / Published online April 17, 2002 RID="*" ID="*" The author is supported by a Heisenberg scholarship of the Deutsche Forschungsgemeinschaft  相似文献   

3.
In 1961, Jan-Erik Roos published a “theorem”, which says that in an [AB4*] abelian category, lim1 vanishes on Mittag–Leffler sequences. See Propositions 1 and 5 in [4]. This is a “theorem” that many people since have known and used. In this article, we outline a counterexample. We construct some strange abelian categories, which are perhaps of some independent interest.?These abelian categories come up naturally in the study of triangulated categories. A much fuller discussion may be found in [3]. Here we provide a brief, self contained, non–technical account. The idea is to make the counterexample easy to read for all the people who have used the result in their work.?In the appendix, Deligne gives another way to look at the counterexample. Oblatum 22-I-2001 & 15-XII-2001?Published online: 1 February 2002  相似文献   

4.
We establish existence and comparison theorems for a class of Hamilton—Jacobi equations. The class of Hamilton—Jacobi equations includes and is broader than those studied in [8] We apply the existence and uniqueness results to characterizing the value functions associated with the optimal control of systems governed by partial differential equations of parabolic type. Accepted 11 May 2001. Online publication 5 October 2001.  相似文献   

5.
We define a generalized notion of mean curvature for regular hypersurfaces in . This enables us to introduce a new class of geometric curvature flows for which we prove enclosure theorems, using methods of Dierkes [D] and Hildebrandt [H]. In particular, we obtain “neck-pinching” results that generalize previous observations by Ecker [E] concerning the classical mean curvature flow. Received: 8 October 2001 / Accepted: 1 March 2002 / Published online: 23 May 2002  相似文献   

6.
 Geometric theories are presented as contraction- and cut-free systems of sequent calculi with mathematical rules following a prescribed rule-scheme that extends the scheme given in Negri and von Plato (1998). Examples include cut-free calculi for Robinson arithmetic and real closed fields. As an immediate consequence of cut elimination, it is shown that if a geometric implication is classically derivable from a geometric theory then it is intuitionistically derivable. Received: 18 April 2001 / Published online: 10 October 2002 Mathematics Subject Classification (2000): 03F05, 18C10, 18B15 Key words or phrases: Cut elimination – Geometric theories – Barr's theorem  相似文献   

7.
In Refs. [J. Math. Anal. Appl. 258:287–308, [2001]; J. Math. Anal. Appl. 256:229–241, [2001]], Yang and Li presented a characterization of preinvex functions and semistrictly preinvex functions under a certain set of conditions. In this note, we show that the same results or even more general ones can be obtained under weaker assumptions; we also give a characterization of strictly preinvex functions under mild conditions. This research was supported by the National Natural Science Foundation of China under Grants 70671064 and 60673177, and the Education Department Foundation of Zhejiang Province Grant 20070306. The authors thank Professor F. Giannessi for valuable comments on the original version of this paper.  相似文献   

8.
We further develop the supersymmetric formalism initiated in [W1] (see also [SjW]). We obtain the optimal mean field bounds at the critical energy for Lyapunov exponents of random walks in random potentials in Z d at weak disorder. This extends some of the results in [W1]. Received: 9 December 1999 / Revised version: 8 May 2000 /?Published online: 15 February 2001  相似文献   

9.
 This article develops a framework of stochastic calculus with respect to a càdlàg finite quadratic variation process. We apply it to the study of a generalization of a semimartingale driven SDE studied by Kurtz, Pardoux and Protter [KPP]. We prove an It?'s formula for functions f(X) of a semimartingale with jumps when f has weak smoothness properties. Examples of X for which this formula is valid are time reversible semimartingales and solutions of [KPP] equations driven by Lévy processes, provided the sum of the absolute values of the jumps, raised to the power 1 + λ, is a.s. finite, where λ takes values between 0 and 1. Received: 1 March 1999 / Revised version: 15 April 2001 / Published online: 11 December 2001  相似文献   

10.
 For any and , let be a generalized Cantor product. The Hausdorff dimension of certain sets concerning are considered. Let be defined as in [11], the exceptional set of values of x for which is not uniformly distributed modulo 1 is also determined. (Received 22 May 2001; in revised form 7 September 2001)  相似文献   

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