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1.
In this paper, we first introduce the concept of Levitin-Polyak well-posedness of a generalized mixed variational inequality in Banach spaces and establish some characterizations of its Levitin-Polyak well-posedness. Under suitable conditions, we prove that the Levitin-Polyak well-posedness of a generalized mixed variational inequality is equivalent to the Levitin-Polyak well-posedness of a corresponding inclusion problem and a corresponding fixed point problem. We also derive some conditions under which a generalized mixed variational inequality in Banach spaces is Levitin-Polyak well-posed. 相似文献
2.
《Optimization》2012,61(1):155-165
In this article, we study well-posedness and stability aspects for vector optimization in terms of minimizing sequences defined using the notion of Henig proper efficiency. We justify the importance of set convergence in the study of well-posedness of vector problems by establishing characterization of well-posedness in terms of upper Hausdorff convergence of a minimizing sequence of sets to the set of Henig proper efficient solutions. Under certain compactness assumptions, a convex vector optimization problem is shown to be well-posed. Finally, the stability of vector optimization is discussed by considering a perturbed problem with the objective function being continuous. By assuming the upper semicontinuity of certain set-valued maps associated with the perturbed problem, we establish the upper semicontinuity of the solution map. 相似文献
3.
M. Bianchi 《Nonlinear Analysis: Theory, Methods & Applications》2010,72(1):460-468
In this paper we introduce some notions of well-posedness for scalar equilibrium problems in complete metric spaces or in Banach spaces. As equilibrium problem is a common extension of optimization, saddle point and variational inequality problems, our definitions originates from the well-posedness concepts already introduced for these problems.We give sufficient conditions for two different kinds of well-posedness and show by means of counterexamples that these have no relationship in the general case. However, together with some additional assumptions, we show via Ekeland’s principle for bifunctions a link between them.Finally we discuss a parametric form of the equilibrium problem and introduce a well-posedness concept for it, which unifies the two different notions of well-posedness introduced in the first part. 相似文献
4.
Alexander J. Zaslavski 《Calculus of Variations and Partial Differential Equations》2001,13(3):265-293
The Tonelli existence theorem in the calculus of variations and its subsequent modifications were established for integrands
f which satisfy convexity and growth conditions. In [27] we considered a class of optimal control problems which is identified
with the corresponding complete metric space of integrands, say . We did not impose any convexity assumptions. The main result in [27] establishes that for a generic integrand the corresponding optimal control problem is well-posed. In this paper we study the set of all integrands for which the corresponding optimal control problem is well-posed. We show that the complement of this set is not only of
the first category but also a -porous set. The main result of the paper is obtained as a realization of a variational principle which can be applied to
various classes of optimization problems.
Received April 15, 2000 / Accepted October 10, 2000 / Published online December 8, 2000 相似文献
5.
《Optimization》2012,61(2)
We consider robustness properties of second-order methods for the sliding mode control of nonlinear ordinary differential equations. A new approach is presented based on the theory of well-posed optimization problems. It is shown that the convergence of the real states of the control system to the ideal one is intimately related to Tykhonov well-posedness of suitably defined dynamic optimization problems. 相似文献
6.
Well-posedness and convexity in vector optimization 总被引:9,自引:0,他引:9
We study a notion of well-posedness in vector optimization through the behaviour of minimizing sequences of sets, defined in terms of Hausdorff set-convergence. We show that the notion of strict efficiency is related to the notion of well-posedness. Using the obtained results we identify a class of well-posed vector optimization problems: the convex problems with compact efficient frontiers. 相似文献
7.
In this paper, we consider Levitin–Polyak well-posedness of parametric generalized equilibrium problems and optimization problems
with generalized equilibrium constraints. Some criteria for these types of well-posedness are derived. In particular, under
certain conditions, we show that generalized Levitin–Polyak well-posedness of a parametric generalized equilibrium problem
is equivalent to the nonemptiness and compactness of its solution set. Finally, for an optimization problem with generalized
equilibrium constraints, we also obtain that, under certain conditions, Levitin–Polyak well-posedness in the generalized sense
is equivalent to the nonemptiness and compactness of its solution set. 相似文献
8.
The aim of this paper is to study Levitin–Polyak (LP in short) well-posedness for set optimization problems. We define the global notions of metrically well-setness and metrically LP well-setness and the pointwise notions of LP well-posedness, strongly DH-well-posedness and strongly B-well-posedness for set optimization problems. Using a scalarization function defined by means of the point-to-set distance, we characterize the LP well-posedness and the metrically well-setness of a set optimization problem through the LP well-posedness and the metrically well-setness of a scalar optimization problem, respectively. 相似文献
9.
Rong Hu Ying-Kang Liu Ya-Ping Fang 《Journal of Fixed Point Theory and Applications》2017,19(4):2209-2223
In this paper, we extend well-posedness notions to the split minimization problem which entails finding a solution of one minimization problem such that its image under a given bounded linear transformation is a solution of another minimization problem. We prove that the split minimization problem in the setting of finite-dimensional spaces is Levitin–Polyak well-posed by perturbations provided that its solution set is nonempty and bounded. We also extend well-posedness notions to the split inclusion problem. We show that the well-posedness of the split convex minimization problem is equivalent to the well-posedness of the equivalent split inclusion problem. 相似文献
10.
In this paper, we consider an extension of well-posedness for a minimization problem to a class of variational–hemivariational
inequalities with perturbations. We establish some metric characterizations for the well-posed variational–hemivariational
inequality and give some conditions under which the variational–hemivariational inequality is strongly well-posed in the generalized
sense. Under some mild conditions, we also prove the equivalence between the well-posedness of variational–hemivariational
inequality and the well-posedness of corresponding inclusion problem. 相似文献
11.
AbstractThis paper investigates geometric properties and well-posedness of a mean curvature flow with volume-dependent forcing. With the class of forcing which bounds the volume of the evolving set away from zero and infinity, we show that a strong version of star-shapedness is preserved over time. More precisely, it is shown that the flow preserves the ρ-reflection property, which corresponds to a quantitative Lipschitz property of the set with respect to the nearest ball. Based on this property we show that the problem is well-posed and its solutions starting with ρ-reflection property become instantly smooth. Lastly, for a model problem, we will discuss the flow’s exponential convergence to the unique equilibrium in Hausdorff topology. For the analysis, we adopt the approach developed by Feldman-Kim to combine viscosity solutions approach and variational method. The main challenge lies in the lack of comparison principle, which accompanies forcing terms that penalize small volume. 相似文献
12.
13.
Hadamard良定性的统一研究 总被引:2,自引:0,他引:2
对一些非线性问题的Hadamard良定性给出一个统一的定理,应用这个定理,可以容易地推出KyFan点,Nash平衡点等的Hadamard良定性。此外,最优化问题和鞍点问题的通用良定性也被研究给出了两个定理。 相似文献
14.
本文研究5阶双周期Kadomtsev-Petviashvili Ⅱ(KP-Ⅱ)方程的局部适定性.具体地,当正则指标s〉-3/4时,本文获得双周期5阶KP-Ⅱ问题在各向异性的Sobolev空间Hs,0(T×T)上的局部适定性.为此,本文充分挖掘KP波所特有的一些对称结构,详细讨论两个波在频率空间垂直方向上分离时相互作用的结果.本文发现,两个波在频率空间上只要不完全重合,就不会发生共振现象.本文的一个重要贡献在于引入一类与Galilie变换可交换的双线性算子,并获得该类双线性算子的L^2有界估计.这些算子的引入可以充分理解KP波的相互作用机制.从而克服之前对Strichartz型估计的依赖,使本文能够很大程度上推进已知的结果. 相似文献
15.
《Nonlinear Analysis: Theory, Methods & Applications》2009,70(12):4585-4603
We introduced and studied the concept of well-posedness to a generalized mixed variational inequality. Some characterizations are given. Under suitable conditions, we prove that the well-posedness of the generalized mixed variational inequality is equivalent to the well-posedness of the corresponding inclusion problem. We also discuss the relations between the well-posedness of the generalized mixed variational inequality and the well-posedness of the corresponding fixed-point problem. Finally, we derive some conditions under which the generalized mixed variational inequality is well-posed. 相似文献
16.
Levitin-Polyak well-posedness of variational inequalities 总被引:1,自引:0,他引:1
In this paper we consider the Levitin-Polyak well-posedness of variational inequalities. We derive a characterization of the Levitin-Polyak well-posedness by considering the size of Levitin-Polyak approximating solution sets of variational inequalities. We also show that the Levitin-Polyak well-posedness of variational inequalities is closely related to the Levitin-Polyak well-posedness of minimization problems and fixed point problems. Finally, we prove that under suitable conditions, the Levitin-Polyak well-posedness of a variational inequality is equivalent to the uniqueness and existence of its solution. 相似文献
17.
In this paper, we first derive several characterizations of the nonemptiness and compactness for the solution set of a convex scalar set-valued optimization problem (with or without cone constraints) in which the decision space is finite-dimensional. The characterizations are expressed in terms of the coercivity of some scalar set-valued maps and the well-posedness of the set-valued optimization problem, respectively. Then we investigate characterizations of the nonemptiness and compactness for the weakly efficient solution set of a convex vector set-valued optimization problem (with or without cone constraints) in which the objective space is a normed space ordered by a nontrivial, closed and convex cone with nonempty interior and the decision space is finite-dimensional. We establish that the nonemptiness and compactness for the weakly efficient solution set of a convex vector set-valued optimization problem (with or without cone constraints) can be exactly characterized as those of a family of linearly scalarized convex set-valued optimization problems and the well-posedness of the original problem. 相似文献
18.
Dontchev Asen L. Gfrerer Helmut Kruger Alexander Y. Outrata Jiří V. 《Set-Valued and Variational Analysis》2020,28(3):451-473
Set-Valued and Variational Analysis - There is a basic paradigm, called here the radius of well-posedness, which quantifies the “distance” from a given well-posed problem to the set of... 相似文献
19.
We revisit the classical theory of multidimensional scalar conservation laws. We reformulate the notion of the classical Kruzkov entropy solutions and study some new properties as well as the well-posedness of the initial value problem with inhomogeneous fluxes and general initial data. We also consider Dirichlet boundary value problems. We put forward a new and transparent definition for solutions and give a simple proof for their well-posedness in domains with smooth boundaries. Finally, we introduce the notion of saturated solutions and show that it is well-posed. 相似文献
20.
本文研究了由目标函数扰动的集值优化问题的有效点集所定义的集值映射的半连续性.讨论了目标函数扰动的集值优化问题在上半连续意义下的稳定性.特别地,在广义适定性条件下,证明了集值优化问题在上半连续意义下的稳定性. 相似文献