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1.
Xiaoqi Yang 《Numerical Functional Analysis & Optimization》2013,34(5-6):621-632
Second-order necessary conditions for inequality and equality constrained C1, 1 optimization problems are derived. A constraint qualification condition which uses the recent generalized second-order directional derivative is employed to obtain these conditions. Various second-order sufficient conditions are given under appropriate conditions on the generalized second-order directional derivative in a neighborhood of a given point. An application of the secondorder conditions to a new class of nonsmooth C1, 1 optimization problems with infinitely many constraints is presented. 相似文献
2.
We present a new second-order directional derivative and study its properties. Using this derivative and the parabolic second-order
derivative, we establish second-order necessary and sufficient optimality conditions for a general scalar optimization problem
by means of the asymptotic and parabolic second-order tangent sets to the feasible set. For the sufficient conditions, the
initial space must be finite dimensional. Then, these conditions are applied to a general vector optimization problem obtaining
second-order optimality conditions that generalize the differentiable case. For this aim, we introduce a scalarization, and
the relationships between the different types of solutions to the vector optimization problem and the scalarized problem are
studied.
This research was partially supported by the Ministerio de Educación y Ciencia (Spain), under projects MTM2006-02629 and Ingenio
Mathematica (i-MATH) CSD2006-00032 (Consolider-Ingenio 2010), 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 valuable comments and suggestions. 相似文献
3.
In this paper, we propose the concept of a second-order composed contingent derivative for set-valued maps, discuss its relationship to the second-order contingent derivative and investigate some of its special properties. By virtue of the second-order composed contingent derivative, we extend the well-known Lagrange multiplier rule and the Kurcyusz–Robinson–Zowe regularity assumption to a constrained set-valued optimization problem in the second-order case. Simultaneously, we also establish some second-order Karush–Kuhn–Tucker necessary and sufficient optimality conditions for a set-valued optimization problem, whose feasible set is determined by a set-valued map, under a generalized second-order Kurcyusz–Robinson–Zowe regularity assumption. 相似文献
4.
引进了一种新的二阶组合切锥, 利用它引进了一种新的二阶组合切导数, 称为二阶组合径向切导数, 并讨论了它的性质及它与二阶组合切导数的关系, 借助二阶径向组合切导数, 分别建立了集值优化取得Benson真有效元的最优性充分和必要条件. 相似文献
5.
Jinchuan Zhou Jingyong Tang Jein-Shan Chen 《Journal of Optimization Theory and Applications》2017,172(3):802-823
In this paper, we study the parabolic second-order directional derivative in the Hadamard sense of a vector-valued function associated with circular cone. The vector-valued function comes from applying a given real-valued function to the spectral decomposition associated with circular cone. In particular, we present the exact formula of second-order tangent set of circular cone by using the parabolic second-order directional derivative of projection operator. In addition, we also deal with the relationship of second-order differentiability between the vector-valued function and the given real-valued function. The results in this paper build fundamental bricks to the characterizations of second-order necessary and sufficient conditions for circular cone optimization problems. 相似文献
6.
Vsevolod I. Ivanov 《Optimization Letters》2010,4(4):597-608
In this paper we obtain second-order optimality conditions of Fritz John and Karush–Kuhn–Tucker types for the problem with
inequality constraints in nonsmooth settings using a new second-order directional derivative of Hadamard type. We derive necessary
and sufficient conditions for a point [`(x)]{\bar x} to be a local minimizer and an isolated local one of order two. In the primal necessary conditions we suppose that all functions
are locally Lipschitz, but in all other conditions the data are locally Lipschitz, regular in the sense of Clarke, Gateaux
differentiable at [`(x)]{\bar x}, and the constraint functions are second-order Hadamard differentiable at [`(x)]{\bar x} in every direction. It is shown by an example that regularity and Gateaux differentiability cannot be removed from the sufficient
conditions. 相似文献
7.
Le Quang Thuy Bui Thi Thanh Nguyen Thi Toan 《Journal of Optimization Theory and Applications》2017,173(2):421-442
Motivated by our recent works on optimality conditions in discrete optimal control problems under a nonconvex cost function, in this paper, we study second-order necessary and sufficient optimality conditions for a discrete optimal control problem with a nonconvex cost function and state-control constraints. By establishing an abstract result on second-order optimality conditions for a mathematical programming problem, we derive second-order necessary and sufficient optimality conditions for a discrete optimal control problem. Using a common critical cone for both the second-order necessary and sufficient optimality conditions, we obtain “no-gap” between second-order optimality conditions. 相似文献
8.
Optimality Conditions in
Differentiable Vector Optimization via Second-Order Tangent Sets 总被引:1,自引:0,他引:1
We provide second-order necessary and sufficient conditions for a point to be an efficient element of a set with respect to
a cone in a normed space, so that there is only a small gap between necessary and sufficient conditions. To this aim, we use
the common second-order tangent set and the asymptotic second-order cone utilized by Penot. As an application we establish
second-order necessary conditions for a point to be a solution of a vector optimization problem with an arbitrary feasible
set and a twice Fréchet differentiable objective function between two normed spaces. We also establish second-order sufficient
conditions when the initial space is finite-dimensional so that there is no gap with necessary conditions. Lagrange multiplier
rules are also given.
This research was partially supported by Ministerio de Ciencia y Tecnología (Spain), Project BFM2003-02194.
Online publication 29 January 2004. 相似文献
9.
We characterize the local upper Lipschitz property of the stationary point mapping and the Karush–Kuhn–Tucker (KKT) mapping for a nonlinear second-order cone programming problem using the graphical derivative criterion. We demonstrate that the second-order sufficient condition and the strict constraint qualification are sufficient for the local upper Lipschitz property of the stationary point mapping and are both sufficient and necessary for the local upper Lipschitz property of the KKT mapping. 相似文献
10.
In this paper, by virtue of an asymptotic second-order contingent derivative and an asymptotic second-order Φ-contingent cone,
differential properties of a class of set-valued maps are investigated and an explicit expression of their asymptotic second-order
contingent derivatives is established. Then, second-order necessary optimality conditions of solutions are obtained for weak
vector variational inequalities. 相似文献
11.
Second-Order Analysis for Control Constrained Optimal Control Problems of Semilinear Elliptic Systems 总被引:2,自引:0,他引:2
J. F. Bonnans 《Applied Mathematics and Optimization》1998,38(3):303-325
This paper presents a second-order analysis for a simple model optimal control problem of a partial differential equation,
namely, a well-posed semilinear elliptic system with constraints on the control variable only. The cost to be minimized is
a standard quadratic functional. Assuming the feasible set to be polyhedric, we state necessary and sufficient second-order
optimality conditions, including a characterization of the quadratic growth condition. Assuming that the second-order sufficient
condition holds, we give a formula for the second-order expansion of the value of the problem as well as the directional derivative
of the optimal control, when the cost function is perturbed. Then we extend the theory of second-order optimality conditions
to the case of vector-valued controls when the feasible set is defined by local and smooth convex constraints. When the space
dimension n is greater than 3, the results are based on a two norms approach, involving spaces L
2
and L
s
, with s>n/2 .
Accepted 27 January 1997 相似文献
12.
Shouyang Wang 《Numerical Functional Analysis & Optimization》2013,34(1-2):237-252
In this paper we study second-order optimality conditions for the multi-objective programming problems with both inequality constraints and equality constraints. Two weak second-order constraint qualifications are introduced, and based on them we derive several second-order necessary conditions for a local weakly efficient solution. Two second-order sufficient conditions are also presented. 相似文献
13.
In this paper, we develop second-order necessary and sufficient optimality conditions for multiobjective optimization problems with both equality and inequality constraints. First, we generalize the Lin fundamental theorem (Ref. 1) to second-order tangent sets; then, based on the above generalized theorem, we derive second-order necessary and sufficient conditions for efficiency. 相似文献
14.
《Optimization》2012,61(3-4):165-185
In this paper, a new generalized second-order directional derivative and a set-valued generalized Hessian are introudced for C1,1 functions in real Banach spaces. It is shown that this set-valued generalized Hessian is single-valued at a point if and only if the function is twice weakly Gãteaux differentiable at the point and that the generalized second-order directional derivative is upper semi-continuous under a regularity condition. Various generalized calculus rules are also given for C1,1 functions. The generalized second-order directional derivative is applied to derive second-order necessary optirnality conditions for mathematical programming problems. 相似文献
15.
No-gap optimality conditions for an optimal control problem with pointwise control-state constraints
An optimal control problem with pointwise mixed constraints of the instationary three-dimensional Navier–Stokes–Voigt equations is considered. We derive second-order optimality conditions and show that there is no gap between second-order necessary optimality conditions and second-order sufficient optimality conditions. In addition, the second-order sufficient optimality conditions for the problem where the objective functional does not contain a Tikhonov regularization term are also discussed. 相似文献
16.
多目标规划局部有效解的二阶条件 总被引:3,自引:0,他引:3
最优性条件的研究一直是多目标规划理论的一个热点,关于有效解的一阶最优性条件的研究,已有大量的文献涌现.可是关于有效解的二阶条件,其研究结果寥寥无几.分析其原因,恐怕主要有两方面.其一,绝大多数多目标优化方法还是基于先将问题标量化,然后借用线性规划或非线性规划中已有的一些成熟的方法来求解,这些方法中的一部分对二阶条件不作任何要求;其二,二阶条件的讨论需要更多的分析工具和更精致的分析 相似文献
17.
《Optimization》2012,61(12):2117-2137
ABSTRACTIn this paper, we study a vector optimization problem (VOP) with both inequality and equality constraints. We suppose that the functions involved are Fréchet differentiable and their Fréchet derivatives are continuous or stable at the point of study. By virtue of a second-order constraint qualification of Abadie type, we provide second-order Karush–Kuhn–Tucker type necessary optimality conditions for the VOP. Moreover, we also obtain second-order sufficient optimality conditions for a kind of strict local efficiency. Both the necessary conditions and the sufficient conditions are shown in equivalent pairs of primal and dual formulations by using theorems of the alternative for the VOP. 相似文献
18.
Attractors and dimension of dissipative lattice systems 总被引:1,自引:0,他引:1
In this paper, by using the argument in [Q.F. Ma, S.H. Wang, C.K. Zhong, Necessary and sufficient conditions for the existence of global attractor for semigroup and application, Indiana Univ. Math. J., 51(6) (2002), 1541-1559.], we prove that the condition given in [S. Zhou, Attractors and approximations for lattice dynamical systems, J. Differential Equations 200 (2004) 342-368.] for the existence of a global attractor for the semigroup associated with general lattice systems on a discrete Hilbert space is a sufficient and necessary condition. As an application, we consider the existence of a global attractor for a second-order lattice system in a discrete weighted space containing all bounded sequences. Finally, we show that the global attractor for first-order and partly dissipative lattice systems corresponding to (partly dissipative) reaction-diffusion equations and second-order dissipative lattice systems corresponding to the strongly damped wave equations have finite fractal dimension if the derivative of the nonlinear term is small at the origin. 相似文献
19.
References 1–4 develop second-order sufficient conditions for local minima of optimal control problems with state and control constraints. These second-order conditions tighten the gap between necessary and sufficient conditions by evaluating a positive-definiteness criterion on the tangent space of the active constraints. The purpose of this paper is twofold. First, we extend the methods in Refs. 3, 4 and include general boundary conditions. Then, we relate the approach to the two-norm approach developed in Ref. 5. A direct sufficiency criterion is based on a quadratic function that satisfies a Hamilton-Jacobi inequality. A specific form of such a function is obtained by applying the second-order sufficient conditions to a parametric optimization problem. The resulting second-order positive-definiteness conditions can be verified by solving Riccati equations.The authors wish to thank K. Malanowski for helpful discussions. 相似文献
20.
Ahmed Taa 《Journal of Global Optimization》2011,50(2):271-291
This paper investigates second-order optimality conditions for general multiobjective optimization problems with constraint
set-valued mappings and an arbitrary constraint set in Banach spaces. Without differentiability nor convexity on the data
and with a metric regularity assumption the second-order necessary conditions for weakly efficient solutions are given in
the primal form. Under some additional assumptions and with the help of Robinson -Ursescu open mapping theorem we obtain dual
second-order necessary optimality conditions in terms of Lagrange-Kuhn-Tucker multipliers. Also, the second-order sufficient
conditions are established whenever the decision space is finite dimensional. To this aim, we use the second-order projective
derivatives associated to the second-order projective tangent sets to the graphs introduced by Penot. From the results obtained
in this paper, we deduce and extend, in the special case some known results in scalar optimization and improve substantially
the few results known in vector case. 相似文献