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1.
In this paper, following the method in the proof of the composition duality principle due to Robinson and using some basic
properties of the ε-subdifferential and the conjugate function of a convex function, we establish duality results for an ε-variational inequality problem. Then, we give Fenchel duality results for the ε-optimal solution of an unconstrained convex optimization problem. Moreover, we present an example to illustrate our Fenchel
duality results for the ε-optimal solutions.
The authors thank the referees for valuable suggestions and comments. This work was supported by Grant No. R01-2003-000-10825-0
from the Basic Research Program of KOSEF. 相似文献
2.
T. Q. Son J. J. Strodiot V. H. Nguyen 《Journal of Optimization Theory and Applications》2009,141(2):389-409
In this paper, ε-optimality conditions are given for a nonconvex programming problem which has an infinite number of constraints. The objective
function and the constraint functions are supposed to be locally Lipschitz on a Banach space. In a first part, we introduce
the concept of regular ε-solution and propose a generalization of the Karush-Kuhn-Tucker conditions. These conditions are up to ε and are obtained by weakening the classical complementarity conditions. Furthermore, they are satisfied without assuming
any constraint qualification. Then, we prove that these conditions are also sufficient for ε-optimality when the constraints are convex and the objective function is ε-semiconvex. In a second part, we define quasisaddlepoints associated with an ε-Lagrangian functional and we investigate their relationships with the generalized KKT conditions. In particular, we formulate
a Wolfe-type dual problem which allows us to present ε-duality theorems and relationships between the KKT conditions and regular ε-solutions for the dual. Finally, we apply these results to two important infinite programming problems: the cone-constrained
convex problem and the semidefinite programming problem. 相似文献
3.
A. Nowakowski 《Journal of Optimization Theory and Applications》2008,138(1):85-93
In this note, we develop a dynamic programming approach for an ε-optimal control problem of Bolza. We prove that each Lipschitz continuous function satisfying the Hamilton-Jacobi inequality
(less than zero and greater than −ε) is an ε-value function. 相似文献
4.
In this paper, approximate solutions of vector optimization problems are analyzed via a metrically consistent ε-efficient concept. Several properties of the ε-efficient set are studied. By scalarization, necessary and sufficient conditions for approximate solutions of convex and
nonconvex vector optimization problems are provided; a characterization is obtained via generalized Chebyshev norms, attaining
the same precision in the vector problem as in the scalarization.
This research was partially supported by the Ministerio de Educación y Ciencia (Spain), Project MTM2006-02629 and by the Consejería
de Educación de la Junta de Castilla y León (Spain), Project VA027B06. The authors are grateful to the anonymous referees
for helpful comments and suggestions. 相似文献
5.
In this paper,by a scalarization method,the lower semicontinuity of the solution mappings to two kinds of parametric generalized vector equilibrium problems involving set-valued mappings is established under new assumptions which are weaker than the C-strict monotonicity.These results extend the corresponding ones.Some examples are given to illustrate our results. 相似文献
6.
In this note, we prove that the convergence results for vector optimization problems with equilibrium constraints presented
in Wu and Cheng (J. Optim. Theory Appl. 125, 453–472, 2005) are not correct. Actually, we show that results of this type cannot be established at all. This is due to the possible lack,
even under nice assumptions, of lower convergence of the solution map for equilibrium problems, already deeply investigated
in Loridan and Morgan (Optimization 20, 819–836, 1989) and Lignola and Morgan (J. Optim. Theory Appl. 93, 575–596, 1997). 相似文献
7.
D. P. Bertsekas 《Journal of Optimization Theory and Applications》2008,139(2):209-225
We consider the problem
where x
i
are multidimensional subvectors of x, f
i
are convex functions, and S is a subspace. Monotropic programming, extensively studied by Rockafellar, is the special case where the subvectors x
i
are the scalar components of x. We show a strong duality result that parallels Rockafellar’s result for monotropic programming, and contains other known
and new results as special cases. The proof is based on the use of ε-subdifferentials and the ε-descent method, which is used here as an analytical vehicle.
Work partially supported by the National Science Foundation Grant No. CCR-9731273. 相似文献
8.
D. Dryanov 《Constructive Approximation》2009,30(1):137-153
Kolmogorov ε-entropy of a compact set in a metric space measures its metric massivity and thus replaces its dimension which is usually infinite. The notion quantifies the compactness property of sets in metric
spaces, and it is widely applied in pure and applied mathematics. The ε-entropy of a compact set is the most economic quantity of information that permits a recovery of elements of this set with
accuracy ε. In the present article we study the problem of asymptotic behavior of the ε-entropy for uniformly bounded classes of convex functions in L
p
-metric proposed by A.I. Shnirelman. The asymptotic of the Kolmogorov ε-entropy for the compact metric space of convex and uniformly bounded functions equipped with L
p
-metric is ε
−1/2, ε→0+.
相似文献
9.
The numerical approximation of nonlinear partial differential equations requires the computation of large nonlinear systems,
that are typically solved by iterative schemes. At each step of the iterative process, a large and sparse linear system has
to be solved, and the amount of time elapsed per step grows with the dimensions of the problem. As a consequence, the convergence
rate may become very slow, requiring massive cpu-time to compute the solution. In all such cases, it is important to improve
the rate of convergence of the iterative scheme. This can be achieved, for instance, by vector extrapolation methods. In this
work, we apply some vector extrapolation methods to the electronic device simulation to improve the rate of convergence of
the family of Gummel decoupling algorithms. Furthermore, a different approach to the topological ε-algorithm is proposed and preliminary results are presented. 相似文献
10.
研究锥伪单调、锥拟凸和上锥连续映射在某种强制性条件下的向量均衡问题解集的特征,建立强制性条件与向量均衡问题解集的关系,得到对偶向量均衡问题局部解集含于向量均衡问题解集的性质和向量均衡问题解集的非空性条件,给出在锥伪单调、锥拟凸和上锥连续映射条件下向量均衡问题解集的非空有界性与强制性条件的等价性. 相似文献
11.
A. Jaśkiewicz 《Journal of Optimization Theory and Applications》2009,141(2):321-347
Zero-sum ergodic semi-Markov games with weakly continuous transition probabilities and lower semicontinuous, possibly unbounded,
payoff functions are studied. Two payoff criteria are considered: the ratio average and the time average. The main result
concerns the existence of a lower semicontinuous solution to the optimality equation and its proof is based on a fixed-point
argument. Moreover, it is shown that the ratio average as well as the time average payoff stochastic games have the same value.
In addition, one player possesses an ε-optimal stationary strategy (ε>0), whereas the other has an optimal stationary strategy.
A. Jaśkiewicz is on leave from Institute of Mathematics and Computer Science, Wrocław University of Technology.
This work is supported by MNiSW Grant 1 P03A 01030. 相似文献
12.
Yu Han 《Numerical Functional Analysis & Optimization》2016,37(11):1420-1437
In this article, stability results concerning the lower semicontinuity and the Hausdorff upper semicontinuity of the solution mappings to parametric generalized vector equilibrium problems with neither the monotonicity of mappings nor any information of the solution mappings are established by using scalarization methods and a new density result. 相似文献
13.
14.
Lai-Jiu Lin Qamrul Hasan Ansari Yu-Jen Huang 《Mathematical Methods of Operations Research》2007,65(1):85-98
In this paper, we consider more general forms of generalized vector quasi-equilibrium problems for multivalued maps which
include many known vector quasi-equilibrium problems and generalized vector quasi-variational inequality problems as special
cases. We establish some existence results for solutions of these problems under pseudomonotonicity and u-hemicontinuity/ℓ-hemicontinuity assumptions.
相似文献
15.
WU YunanInstitute of Policy Management Chinese Academy of Sciences Beijing China 《中国科学A辑(英文版)》2004,47(1):65-71
The concept of vector optimization problems with equilibrium constraints (VOPEC) is introduced. By using the continuity results of the approximate solution set to the equilibrium problem, we obtain the same results of the marginal map and the approximate value in VOPEC (e) for vector-valued mapping. 相似文献
16.
The purpose of this paper is to study systems of vector equilibrium problems. We establish some existence theorems for systems
of vector equilibrium problems by using (S)+-conditions and Kakutani–Fan–Glicksberg fixed point theorem
*This work was supported by the Kyungnam University Research Fund 2004 相似文献
17.
Let X and Y be Hausdorff topological vector spaces, K a nonempty, closed, and convex subset of X, C : K → 2Y a point-to-set mapping such that for any χ ε K, C(χ) is a pointed, closed, and convex cone in Y and int C(χ) ≠ 0. Given a mapping g : K → K and a vector valued bifunction f : K × K → Y, we consider the implicit vector equilibrium problem (IVEP) of finding χ* ε K such that f g(χ*), y) -int C(χ) for all y ε K. This problem generalizes the (scalar) implicit equilibrium problem and implicit variational inequality problem. We propose the dual of the implicit vector equilibrium problem (DIVEP) and establish the equivalence between (IVEP) and (DIVEP) under certain assumptions. Also, we give characterizations of the set of solutions for (IVP) in case of nonmonotonicity, weak C-pseudomonotonicity, C-pseudomonotonicity, and strict C-pseudomonotonicity, respectively. Under these assumptions, we conclude that the sets of solutions are nonempty, closed, and convex. Finally, we give some applications of (IVEP) to vector variational inequality problems and vector optimization problems. 相似文献
18.
Convergence of stochastic search algorithms to finite size pareto set approximations 总被引:1,自引:0,他引:1
Oliver Schütze Marco Laumanns Carlos A. Coello Coello Michael Dellnitz El-Ghazali Talbi 《Journal of Global Optimization》2008,41(4):559-577
In this work we investigate the convergence of stochastic search algorithms toward the Pareto set of continuous multi-objective
optimization problems. The focus is on obtaining a finite approximation that should capture the entire solution set in a suitable
sense, which will be defined using the concept of ε-dominance. Under mild assumptions about the process to generate new candidate solutions, the limit approximation set will
be determined entirely by the archiving strategy. We propose and analyse two different archiving strategies which lead to
a different limit behavior of the algorithms, yielding bounds on the obtained approximation quality as well as on the cardinality
of the resulting Pareto set approximation.
相似文献
19.
Several authors have been intersted in optimality conditions in ε-programming and in ε-minimas problem (see, for example,
the references [11,16]).
In this paper, we present some results for approximating ε-programming and ε-minimax problems with penalty techniques. From
a computational point of view, such results may be used in order to improve algorithms within a given level of accuracy. 相似文献
20.
Tadeusz Antczak 《Journal of Global Optimization》2009,43(1):111-140
This paper represents the second part of a study concerning the so-called G-multiobjective programming. A new approach to duality in differentiable vector optimization problems is presented. The techniques
used are based on the results established in the paper: On G-invex multiobjective programming. Part I. Optimality by T.Antczak. In this work, we use a generalization of convexity, namely G-invexity, to prove new duality results for nonlinear differentiable multiobjective programming problems. For such vector
optimization problems, a number of new vector duality problems is introduced. The so-called G-Mond–Weir, G-Wolfe and G-mixed dual vector problems to the primal one are defined. Furthermore, various so-called G-duality theorems are proved between the considered differentiable multiobjective programming problem and its nonconvex vector
G-dual problems. Some previous duality results for differentiable multiobjective programming problems turn out to be special
cases of the results described in the paper. 相似文献