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1.
《Set-Valued Analysis》2008,16(2-3):199-227
The paper contains two groups of results. The first are criteria for calmness/subregularity for set-valued mappings between
finite-dimensional spaces. We give a new sufficient condition whose subregularity part has the same form as the coderivative
criterion for “full” metric regularity but involves a different type of coderivative which is introduced in the paper. We
also show that the condition is necessary for mappings with convex graphs. The second group of results deals with the basic
calculus rules of nonsmooth subdifferential calculus. For each of the rules we state two qualification conditions: one in
terms of calmness/subregularity of certain set-valued mappings and the other as a metric estimate (not necessarily directly
associated with aforementioned calmness/subregularity property). The conditions are shown to be weaker than the standard Mordukhovich–Rockafellar
subdifferential qualification condition; in particular they cover the cases of convex polyhedral set-valued mappings and,
more generally, mappings with semi-linear graphs. Relative strength of the conditions is thoroughly analyzed. We also show,
for each of the calculus rules, that the standard qualification conditions are equivalent to “full” metric regularity of precisely
the same mappings that are involved in the subregularity version of our calmness/subregularity condition.
The research of Jiří V. Outrata was supported by the grant A 107 5402 of the Grant Agency of the Academy of Sciences of the
Czech Republic. 相似文献
2.
Nguyen Thi Van Hang 《Numerical Functional Analysis & Optimization》2013,34(3):364-386
This article shows that finite-dimensional multiplier rules, which are based on the limiting subdifferential, can be proved by Ekeland's variational principle and some basic calculus tools of the generalized differentiation theory introduced by B. S. Mordukhovich. Consequences of a limiting constraint qualification, which yields the normal form of the multiplier rules, stability and calmness of optimization problems, are investigated in detail. 相似文献
3.
Michel H. Geoffroy 《Journal of Mathematical Analysis and Applications》2006,313(2):689-699
We present an iterative procedure for solving, in finite dimensions, generalized equations of the form
(∗) 相似文献
4.
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
5.
The main aspect of the paper consists in the application of a particular theorem of separation between two sets to the image
associated with a constrained extremum problem. In the image space, the two sets are a convex cone, which depends on the constraints
(equalities or inequalities) of the given problem, and its image. In this way, a condition for the existence of a regular
saddle point (i.e., a sufficient optimality condition) is obtained. This regularity condition is compared with those existing
in the literature. 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.
The paper is devoted to the analysis of the calmness property for constraint set mappings. After some general characterizations,
specific results are obtained for various types of constraints, e.g., one single nonsmooth inequality, differentiable constraints
modeled by polyhedral sets, finitely and infinitely many differentiable inequalities. The obtained conditions enable the detection
of calmness in a number of situations, where the standard criteria (via polyhedrality or the Aubin property) do not work.
Their application in the framework of generalized differential calculus is explained and illustrated by examples associated
with optimization and stability issues in connection with nonlinear complementarity problems or continuity of the value-at-risk.
This research was supported by the DFG Research center Matheon Mathematics for key technologies in Berlin
Support by grant IAA1030405 of the Grant Agency of the Academy of Sciences of the Czech Republic is acknowledged 相似文献
8.
We establish both necessary and sufficient optimality conditions for weak efficiency and firm efficiency by using Hadamard
directional derivatives and scalarizing the multiobjective problem under consideration via signed distances. For the first-order
conditions, the data of the problem need not even be continuous; for the second-order conditions, we assume only that the
first-order derivatives of the data are calm. We include examples showing the advantages of our results over some recent papers
in the literature.
This work was partially supported by the National Basic Research Program in Natural Sciences of Vietnam. The authors are indebted
to Professor P.L. Yu and two anonymous referees for many valuable remarks, which helped improving the previous version of
the paper. 相似文献
9.
Constraint qualifications in terms of approximate Jacobians are investigated for a nonsmooth constrained optimization problem, in which the involved functions are continuous but not necessarily locally Lipschitz. New constraint qualifications in terms of approximate Jacobians, weaker than the generalized Robinson constraint qualification (GRCQ) in Jeyakumar and Yen [V. Jeyakumar, N.D. Yen, Solution stability of nonsmooth continuous systems with applications to cone-constrained optimization, SIAM J. Optim. 14 5 (2004) 1106-1127], are introduced and some examples are provided to show the utility of constrained qualifications introduced. Since the calmness condition is regarded as the basic condition for optimality conditions, the relationships between the constraint qualifications proposed and the calmness of solution mapping are also studied. 相似文献
10.
Tim Hoheisel Ji?í V. Outrata 《Nonlinear Analysis: Theory, Methods & Applications》2010,72(5):2514-2526
A mathematical program with vanishing constraints (MPVC) is a constrained optimization problem arising in certain engineering applications. The feasible set has a complicated structure so that the most familiar constraint qualifications are usually violated. This, in turn, implies that standard penalty functions are typically non-exact for MPVCs. We therefore develop a new MPVC-tailored penalty function which is shown to be exact under reasonable assumptions. This new penalty function can then be used to derive (or recover) suitable optimality conditions for MPVCs. 相似文献