共查询到20条相似文献,搜索用时 31 毫秒
1.
Thai Doan Chuong 《Nonlinear Analysis: Theory, Methods & Applications》2011,74(17):6256-6264
The aim of this work is twofold. First, we use the advanced tools of modern variational analysis and generalized differentiation to study the Lipschitz-like property of an implicit multifunction. More explicitly, new sufficient conditions in terms of the Fréchet coderivative and the normal/Mordukhovich coderivative of parametric multifunctions for this implicit multifunction to have the Lipschitz-like property at a given point are established. Then we derive sufficient conditions ensuring the Lipschitz-like property of an efficient solution map in parametric vector optimization problems by employing the above implicit multifunction results. 相似文献
2.
Our paper deals with the interrelation of optimization methods and Lipschitz stability of multifunctions in arbitrary Banach spaces. Roughly speaking, we show that linear convergence of several first order methods and Lipschitz stability mean the same. Particularly, we characterize calmness and the Aubin property by uniformly (with respect to certain starting points) linear convergence of descent methods and approximate projection methods. So we obtain, e.g., solution methods (for solving equations or variational problems) which require calmness only. The relations of these methods to several known basic algorithms are discussed, and errors in the subroutines as well as deformations of the given mappings are permitted. We also recall how such deformations are related to standard algorithms like barrier, penalty or regularization methods in optimization. 相似文献
3.
Roxin Zhang 《Mathematical Programming》2005,102(1):153-166
The main purpose of this paper is to report on our studies of the weak upper Lipschitz and weak -upper Lipschitz continuities of multifunctions. Comparisons with other related Lipschitz-type continuities and calmness are given. When the concept of the weak upper Lipschitz continuities is applied to the special cases of constraint multifunctions, such as ones defined by a systems of equalities and inequalities or by a generalized equation we obtain the equivalent conditions with linear functional error bounds. Some results on the perturbation and penalty issues in parametric optimization problems are obtained under weak upper Lipschitz continuity assumptions on the constraint multifunctions. We also discuss the weak -upper Lipschitz continuity of a inverse subdifferential.Mathematics Subject Classification (2000): 49J52, 49J53, 90C25Acknowledgement The author thanks the associate editor and the referees for their helpful suggestions and comments. 相似文献
4.
The Abadie CQ (ACQ) for convex inequality systems is a fundamental notion in optimization and approximation theory. In terms of the contingent cone and tangent derivative, we extend the Abadie CQ to more general convex multifunction cases and introduce the strong ACQ for both multifunctions and inequality systems. Some seemly unrelated notions are unified by the new ACQ and strong ACQ. Relationships among ACQ, strong ACQ, basic constraint qualification (BCQ) and strong BCQ are discussed. Using the strong ACQ, we study calmness of a closed and convex multifunction between two Banach spaces and, different from many existing dual conditions for calmness, establish several primal characterizations of calmness. As applications, some primal characterizations for error bounds and linear regularity are established; in particular, some existing results are improved. 相似文献
5.
Adam B. Levy 《Set-Valued Analysis》2006,14(3):249-261
Calmness of multifunctions is a well-studied concept of generalized continuity in which single-valued selections from the image sets of the multifunction exhibit a restricted type of local Lipschitz continuity where the base point is fixed as one point of comparison. Generalized continuity properties of multifunctions like calmness can be applied to convergence analysis when the multifunction appropriately represents the iterates generated by some algorithm. Since it involves an essentially linear relationship between input and output, calmness gives essentially linear convergence results when it is applied directly to convergence analysis. We introduce a new continuity concept called ‘supercalmness’ where arbitrarily small calmness constants can be obtained near the base point, which leads to essentially superlinear convergence results. We also explore partial supercalmness and use a well-known generalized derivative to characterize both when a multifunction is supercalm and when it is partially supercalm. To illustrate the value of such characterizations, we explore in detail a new example of a general primal sequential quadratic programming method for nonlinear programming and obtain verifiable conditions to ensure convergence at a superlinear rate. 相似文献
6.
本文研究了集值映射向量优化问题的锥弱有效解的镇定性和稳定性,我们引进了集值映射向量优化问题的镇定性和稳定性的定义,并证明了集值映射向量优化问题的镇定性和稳定性的一些主要定理. 相似文献
7.
In this paper, some necessary and sufficient optimality conditions for the weakly efficient solutions of vector optimization
problems (VOP) with finite equality and inequality constraints are shown by using two kinds of constraints qualifications
in terms of the MP subdifferential due to Ye. A partial calmness and a penalized problem for the (VOP) are introduced and
then the equivalence between the weakly efficient solution of the (VOP) and the local minimum solution of its penalized problem
is proved under the assumption of partial calmness.
This work was supported by the National Natural Science Foundation of China (10671135), the Specialized Research Fund for
the Doctoral Program of Higher Education (20060610005) and the National Natural Science Foundation of Sichuan Province (07ZA123).
The authors thank Professor P.M. Pardalos and the referees for comments and suggestions. 相似文献
8.
In this paper, by virtue of two intermediate derivative-like multifunctions, which depend on an element in the intermediate
space, some exact calculus rules are obtained for calculating the derivatives of the composition of two set-valued maps. Similar
rules are displayed for sums. Moreover, by using these calculus rules, the solution map of a parametrized variational inequality
and the variations of the feasible set of a parametrized mathematical programming problem are studied.
This research was partially supported by the National Natural Science Foundation of China (Grant numbers: 10871216 and 60574073). 相似文献
9.
René Henrion Alexander Y. Kruger Jiří V. Outrata 《Journal of Optimization Theory and Applications》2013,159(3):681-697
The paper concerns the computation of the graphical derivative and the regular (Fréchet) coderivative of the solution map to a class of generalized equations, where the multivalued term amounts to the regular normal cone to a (possibly nonconvex) set given by C 2 inequalities. Instead of the linear independence qualification condition, standardly used in this context, one assumes a combination of the Mangasarian–Fromovitz and the constant rank qualification conditions. Based on the obtained generalized derivatives, new optimality conditions for a class of mathematical programs with equilibrium constraints are derived, and a workable characterization of the isolated calmness of the considered solution map is provided. 相似文献
10.
M. J. Cánovas M. A. López J. Parra F. J. Toledo 《Set-Valued and Variational Analysis》2014,22(2):375-389
In this paper we deal with parameterized linear inequality systems in the n-dimensional Euclidean space, whose coefficients depend continuosly on an index ranging in a compact Hausdorff space. The paper is developed in two different parametric settings: the one of only right-hand-side perturbations of the linear system, and that in which both sides of the system can be perturbed. Appealing to the backgrounds on the calmness property, and exploiting the specifics of the current linear structure, we derive different characterizations of the calmness of the feasible set mapping, and provide an operative expresion for the calmness modulus when confined to finite systems. In the paper, the role played by the Abadie constraint qualification in relation to calmness is clarified, and illustrated by different examples. We point out that this approach has the virtue of tackling the calmness property exclusively in terms of the system’s data. 相似文献
11.
We obtain some point-based sufficient conditions for the metric regularity in Robinson’s sense of implicit multifunctions in a finite-dimensional setting. The new implicit function theorem (which is very different from the preceding results of Ledyaev and Zhu [Yu.S. Ledyaev, Q.J. Zhu, Implicit multifunctions theorems, Set-Valued Anal. 7 (1999) 209–238], Ngai and Théra [H.V. Ngai, M. Théra, Error bounds and implicit multifunction theorem in smooth Banach spaces and applications to optimization, Set-Valued Anal. 12 (2004) 195–223], Lee, Tam and Yen [G.M. Lee, N.N. Tam, N.D. Yen, Normal coderivative for multifunctions and implicit function theorems, J. Math. Anal. Appl. 338 (2008) 11–22]) can be used for analyzing parametric constraint systems as well as parametric variational systems. Our main tools are the concept of normal coderivative due to Mordukhovich and the corresponding theory of generalized differentiation. 相似文献
12.
Using variational analysis, we study the vector optimization problems with objectives being closed multifunctions on Banach spaces or in Asplund spaces. In terms of the coderivatives and normal cones, we present Fermat’s rules as necessary or sufficient conditions for a super efficient solution of the above problems. 相似文献
13.
We consider parametric equilibrium problems in metric spaces. Sufficient conditions for the Hölder calmness of solutions are established. We also study the Hölder well-posedness for equilibrium problems in metric spaces. 相似文献
14.
Thai Doan Chuong 《Optimization Letters》2013,7(6):1087-1117
The normal subdifferential of a set-valued mapping with values in a partially ordered Banach space has been recently introduced in Bao and Mordukhovich (Control Cyber 36:531–562, 2007), by using the Mordukhovich coderivative of the associated epigraphical multifunction, which has proven to be useful in deriving necessary conditions for super efficient points of vector optimization problems. In this paper, we establish new formulae for computing and/or estimating the normal subdifferential of the efficient point multifunctions of parametric vector optimization problems. These formulae will be presented in a broad class of conventional vgector optimization problems with the presence of geometric, operator, equilibrium, and (finite and infinite) functional constraints. 相似文献
15.
《Optimization》2012,61(6):1131-1156
ABSTRACTThis paper is concerned with the strong calmness of the KKT solution mapping for a class of canonically perturbed conic programming, which plays a central role in achieving fast convergence under situations when the Lagrange multiplier associated to a solution of these conic optimization problems is not unique. We show that the strong calmness of the KKT solution mapping is equivalent to a local error bound for the solutions to the perturbed KKT system, and is also equivalent to the pseudo-isolated calmness of the stationary point mapping along with the calmness of the multiplier set mapping at the corresponding reference point. Sufficient conditions are also provided for the strong calmness by establishing the pseudo-isolated calmness of the stationary point mapping in terms of the noncriticality of the associated multiplier, and the calmness of the multiplier set mapping in terms of a relative interior condition for the multiplier set. These results cover and extend the existing ones in Hager and Gowda [Stability in the presence of degeneracy and error estimation. Math Program. 1999;85:181–192]; Izmailov and Solodov [Stabilized SQP revisited. Math Program. 2012;133:93–120] for nonlinear programming and in Cui et al. [On the asymptotic superlinear convergence of the augmented Lagrangian method for semidefinite programming with multiple solutions. 2016, arXiv: 1610.00875v1]; Zhang and Zhang [Critical multipliers in semidefinite programming. 2018, arXiv: 1801.02218v1] for semidefinite programming. 相似文献
16.
Francisco J. Aragón Artacho Boris S. Mordukhovich 《Journal of Global Optimization》2011,50(1):145-167
This paper mainly concerns the study of a large class of variational systems governed by parametric generalized equations,
which encompass variational and hemivariational inequalities, complementarity problems, first-order optimality conditions,
and other optimization-related models important for optimization theory and applications. An efficient approach to these issues
has been developed in our preceding work (Aragón Artacho and Mordukhovich in Nonlinear Anal 72:1149–1170, 2010) establishing qualitative and quantitative relationships between conventional metric regularity/subregularity and Lipschitzian/calmness
properties in the framework of parametric generalized equations in arbitrary Banach spaces. This paper provides, on one hand,
significant extensions of the major results in op.cit. to partial metric regularity and to the new hemiregularity property.
On the other hand, we establish enhanced relationships between certain strong counterparts of metric regularity/hemiregularity
and single-valued Lipschitzian localizations. The results obtained are new in both finite-dimensional and infinite-dimensional
settings. 相似文献
17.
Jean-Pierre Aubin 《Journal of Evolution Equations》2006,6(3):419-432
The purpose of this paper is to characterize by means of viability tools the pseudo-lipschitzianity property of a set-valued
map F in a neighborhood of a point of its graph in terms of derivatives of this set-valued map F in a neighborhood of a point of its graph, instead of using the transposes of the derivatives. On the way, we relate these
properties to the calmness index of a set-valued map, an extensions of Clarke’s calmness of a function, as well as Doyen’s
Lipschitz kernel of a set-valued map, which is the largest Lipschitz submap. 相似文献
18.
19.
A. Uderzo 《Journal of Optimization Theory and Applications》2014,162(1):47-78
In this paper, two properties of recognized interest in variational analysis, known as Lipschitz lower semicontinuity and calmness, are studied with reference to a general class of variational systems, i.e. to solution mappings to parameterized generalized equations. In the consideration of the metric nature of such properties, some related sufficient conditions are established, which are expressed via nondegeneracy conditions on derivative-like objects appropriate for a metric space analysis. For certain classes of generalized equations in Asplund spaces, it is shown how such conditions can be formulated by using the Fréchet coderivative of the field and the derivative of the base. Applications to the stability analysis of parametric constrained optimization problems are proposed. 相似文献
20.
The aim of this paper is to investigate the convergence properties for Mordukhovich’s coderivative of the solution map of
the sample average approximation (SAA) problem for a parametric stochastic generalized equation. It is demonstrated that,
under suitable conditions, both the cosmic deviation and the ρ-deviation between the coderivative of the solution mapping to SAA problem and that of the solution mapping to the parametric
stochastic generalized equation converge almost surely to zero as the sample size tends to infinity. Moreover, the exponential
convergence rate of coderivatives of the solution maps to the SAA parametric generalized equations is established. The results
are used to develop sufficient conditions for the consistency of the Lipschitz-like property of the solution map of SAA problem
and the consistency of stationary points of the SAA estimator for a stochastic mathematical program with complementarity constraints. 相似文献