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1.
《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
2.
First and second-order approximations are used to establish both necessary and sufficient optimality conditions for local
weak efficiency and local firm efficiency in nonsmooth set-constrained vector problems. Even continuity and relaxed convexity
assumptions are not imposed. Compactness conditions are also relaxed. Examples are provided to show advantages of the presented
results over recent existing ones.
This work was supported in part by the National Basic Research Program in Natural Sciences of Vietnam. 相似文献
3.
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
4.
Nikolai P. Osmolovskii 《Journal of Mathematical Analysis and Applications》2018,457(2):1613-1633
One of the first steps towards necessary second-order optimality conditions in problems with constraints was taken by Dubovitskii and Milyutin in 1965. They offered a scheme that was very effective in smooth optimization problems, but seemed to be not suitable for applications in problems with pointwise control constraints. In this article we consider a modification of the Dubovitskii–Milyutin scheme, which allows to derive necessary second-order conditions for a weak local minimum in optimal control problems with a finite number of endpoint constraints of equality and inequality type and with pointwise control constraints of inequality type given by smooth functions. Assuming that the gradients of active control constraints are linearly independent, we provide rather straightforward proof of these conditions for a measurable and essentially bounded optimal control. 相似文献
5.
Second-order necessary optimality conditions play an important role in optimization theory. This is explained by the fact that most numerical optimization algorithms reduce to finding stationary points satisfying first-order necessary optimality conditions. As a rule, optimization problems, especially the high dimensional ones, have a lot of stationary points so one has to use second-order necessary optimality conditions to exclude nonoptimal points. These conditions are closely related to second-order constraint qualifications, which guarantee the validity of second-order necessary optimality conditions. In this paper, strong and weak second-order necessary optimality conditions are considered and their validity proved under so-called critical regularity condition at local minimizers. 相似文献
6.
In this paper, we propose several second-order derivatives for set-valued maps and discuss their properties. By using these derivatives, we obtain second-order necessary optimality conditions for strict efficiency of a set-valued optimization problem with inclusion constraints in real normed spaces. We also establish second-order sufficient optimality conditions for strict efficiency of the set-valued optimization problem in finite-dimensional normed spaces. As applications, we investigate second-order sufficient and necessary optimality conditions for a strict local efficient solution of order two of a nonsmooth vector optimization problem with an abstract set and a functional constraint. 相似文献
7.
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
8.
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
9.
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
10.
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
11.
Dušan Bedna?ík Karel Pastor 《Nonlinear Analysis: Theory, Methods & Applications》2011,74(4):1372-1382
In this paper we study a multiobjective optimization problem with inequality constraints on finite dimensional spaces. A second-order necessary condition for local weak efficiency is proved under strict differentiability assumptions. We also establish a second-order sufficient condition for local firm efficiency of order 2 under ?-stability assumptions. In this way we generalize some corresponding results obtained by P.Q. Khanh and N.D. Tuan, and by the authors. 相似文献
12.
在赋范空间中给出了集值映射的二阶切集的概念,利用二阶切集,定义了集值映射的二阶切导数。然后,获得了集值向量优化问题弱极小元的两个二阶最优性必要条件。 相似文献
13.
Matheus J. Lazo 《Optimization》2014,63(8):1157-1165
Fractional operators play an important role in modelling nonlocal phenomena and problems involving coarse-grained and fractal spaces. The fractional calculus of variations with functionals depending on derivatives and/or integrals of noninteger order is a rather recent subject that is currently in fast development due to its applications in physics and other sciences. In the last decade, several approaches to fractional variational calculus were proposed by using different notions of fractional derivatives and integrals. Although the literature of the fractional calculus of variations is already vast, much remains to be done in obtaining necessary and sufficient conditions for the optimization of fractional variational functionals, existence and regularity of solutions. Regarding necessary optimality conditions, all works available in the literature concern the derivation of first-order fractional conditions of Euler–Lagrange type. In this work, we obtain a Legendre second-order necessary optimality condition for weak extremizers of a variational functional that depends on fractional derivatives. 相似文献
14.
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
15.
In this paper, we are concerned with a nonsmooth multiobjective optimization problem with inequality constraints. We introduce a second-order constraint qualification, which is a generalization of the Abadie constraint qualification and derive second-order Kuhn-Tucker type necessary conditions for efficiency under the constraint qualification. Moreover, we give some conditions which ensure the constraint qualification holds. 相似文献
16.
Journal of Optimization Theory and Applications - The main goal of this paper is to give some primal and dual Karush–Kuhn–Tucker second-order necessary conditions for the existence of a... 相似文献
17.
Daniel Hoehener 《Applied Mathematics and Optimization》2013,68(2):219-253
We propose second-order necessary optimality conditions for optimal control problems with very general state and control constraints which hold true under weak regularity assumptions on the data. In particular the pure state constraints are general closed sets, the optimal control is supposed to be merely measurable and the dynamics may be discontinuous in the time variable as well. These results are obtained by an approach based on local perturbations of the reference process by second-order tangent directions. This method allows direct and quite simple proofs. 相似文献
18.
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
19.
In this paper, we discuss the existence and controllability for a class of second-order evolution differential inclusions without compactness in Banach spaces. By applying the technique of weak topology and Glicksberg–Ky Fan fixed point theorem, we prove our main results without the hypotheses of compactness on the operator generated by the linear part and any conditions on the multivalued nonlinearity expressed in terms of measures of noncompactness. Further, we extend our study to existence and controllability of second-order evolution differential inclusions with nonlocal conditions and impulses. Finally, an example is given for the illustration of the obtained theoretical results. 相似文献
20.
A weak Galerkin finite element method for the second order elliptic problems with mixed boundary conditions
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Saqib Hussain Nolisa Malluwawadu Peng Zhu 《Journal of Applied Analysis & Computation》2018,8(5):1452-1463
In this paper, a weak Galerkin finite element method is proposed and analyzed for the second-order elliptic equation with mixed boundary conditions. Optimal order error estimates are established in both discrete $H^1$ norm and the standard $L^2$ norm for the corresponding WG approximations. The numerical experiments are presented to verify the efficiency of the method. 相似文献