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1.
L. T. Tung 《Set-Valued and Variational Analysis》2018,26(3):561-579
In this paper, second-order sensitivity analysis in vector optimization problems is considered. We prove that the efficient solution map and the efficient frontier map of a parameterized vector optimization problem are second-order proto-differentiable under some appropriate qualification conditions. Some sufficient conditions for inner and outer approximation of the second-order proto-derivative are also provided. 相似文献
2.
Second-order sufficient condition and quadratic growth condition play important roles both in sensitivity and stability analysis and in numerical analysis for optimization problems. In this article, we concentrate on the global quadratic growth condition and study its relations with global second-order sufficient conditions for min-max optimization problems with quadratic functions. In general, the global second-order sufficient condition implies the global quadratic growth condition. In the case of two quadratic functions involved, we have the equivalence of the two conditions. 相似文献
3.
Ben Liu 《Numerical Functional Analysis & Optimization》2013,34(1):50-67
In this article, we introduce a second-order modified contingent cone and a second-order modified contingent epiderivative. We discuss some properties of the second-order cone and the epiderivative, respectively. Moreover, a Fritz John type necessary optimality condition is obtained for the set-valued optimization problems with constraints by using the second-order modified contingent epiderivative and an example is proposed to explain the Fritz John type necessary optimality condition. In particular, we obtain a unified second-order sufficient and necessary optimality condition for the set-valued optimization problems with constraints under twice differentiable L-quasi-convex assumption. 相似文献
4.
The main purpose of this paper is to make use of the second-order subdifferential of vector functions to establish necessary and sufficient optimality conditions for vector optimization problems. 相似文献
5.
Kazimierz Malanowski 《Applied Mathematics and Optimization》1990,21(1):1-20
A family of optimization problems in a Hilbert space depending on a vector parameter is considered. It is assumed that the problems have locally isolated local solutions. Both these solutions and the associated Lagrange multipliers are assumed to be locally Lipschitz continuous functions of the parameter. Moreover, the assumption of the type of strong second-order sufficient condition is satisfied.It is shown that the solutions are directionally differentiable functions of the parameter and the directional derivative is characterized. A second-order expansion of the optimal-value function is obtained. The abstract results are applied to state and control constrained optimal control problems for systems described by nonlinear ordinary differential equations with the control appearing linearly. 相似文献
6.
E. R. Avakov A. V. Arutyunov A. F. Izmailov 《Computational Mathematics and Mathematical Physics》2008,48(3):346-353
A new first-order sufficient condition for penalty exactness that includes neither the standard constraint qualification requirement nor the second-order sufficient optimality condition is proposed for optimization problems with equality constraints. 相似文献
7.
In this paper, we study second-order optimality conditions for multiobjective optimization problems. By means of different
second-order tangent sets, various new second-order necessary optimality conditions are obtained in both scalar and vector
optimization. As special cases, we obtain several results found in the literature (see reference list). We present also second-order
sufficient optimality conditions so that there is only a very small gap with the necessary optimality conditions.
The authors thank Professor P.L. Yu and the referees for valuable comments and helpful suggestions. 相似文献
8.
The stabilized version of the sequential quadratic programming algorithm (sSQP) had been developed in order to achieve fast
convergence despite possible degeneracy of constraints of optimization problems, when the Lagrange multipliers associated
to a solution are not unique. Superlinear convergence of sSQP had been previously established under the strong second-order
sufficient condition for optimality (without any constraint qualification assumptions). We prove a stronger superlinear convergence
result than the above, assuming the usual second-order sufficient condition only. In addition, our analysis is carried out
in the more general setting of variational problems, for which we introduce a natural extension of sSQP techniques. In the
process, we also obtain a new error bound for Karush–Kuhn–Tucker systems for variational problems that holds under an appropriate
second-order condition. 相似文献
9.
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
10.
In this paper, we introduce the concept of a generalized second-order composed contingent epiderivative for set-valued maps
and discuss its relationship to the generalized second-order contingent epiderivative. We also investigate some of its properties.
Then, by virtue of the generalized second-order composed contingent epiderivative, we establish a unified second-order sufficient
and necessary optimality condition for set-valued optimization problems, which is a generalization of the corresponding results
in the literature. 相似文献
11.
W. Alt 《Journal of Optimization Theory and Applications》1991,70(3):443-466
This paper considers a class of nonlinear differentiable optimization problems depending on a parameter. We show that, if constraint regularity, a second-order sufficient optimality condition, and a stability condition for the Lagrange multipliers hold, then for sufficiently smooth perturbations of the constraints and the objective function the optimal solutions locally obey a type of Lipschitz condition. The results are applied to finite-dimensional problems, equality constrained problems, and optimal control problems. 相似文献
12.
This paper deals with generalized semi-infinite optimization problems where the (infinite) index set of inequality constraints depends on the state variables and all involved functions are twice continuously differentiable. Necessary and sufficient second-order optimality conditions for such problems are derived under assumptions which imply that the corresponding optimal value function is second-order (parabolically) directionally differentiable and second-order epiregular at the considered point. These sufficient conditions are, in particular, equivalent to the second-order growth condition. 相似文献
13.
X. Q. Yang 《Journal of Optimization Theory and Applications》1997,95(3):729-734
The study of a vector variational inequality has been advanced because it has many applications in vector optimization problems and vector equilibrium flows. In this paper, we discuss relations between a solution of a vector variational inequality and a Pareto solution or a properly efficient solution of a vector optimization problem. We show that a vector variational inequality is a necessary and sufficient optimality condition for an efficient solution of the vector pseudolinear optimization problem. 相似文献
14.
The concept of a cone subarcwise connected set-valued map is introduced. Several examples are given to illustrate that the cone subarcwise connected set-valued map is a proper generalization of the cone arcwise connected set-valued map, as well as the arcwise connected set is a proper generalization of the convex set, respectively. Then, by virtue of the generalized second-order contingent epiderivative, second-order necessary optimality conditions are established for a point pair to be a local global proper efficient element of set-valued optimization problems. When objective function is cone subarcwise connected, a second-order sufficient optimality condition is also obtained for a point pair to be a global proper efficient element of set-valued optimization problems. 相似文献
15.
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
16.
Francesco Ludovici Ira Neitzel Winnifried Wollner 《Journal of Optimization Theory and Applications》2018,178(2):317-348
We consider the finite element discretization of semilinear parabolic optimization problems subject to pointwise in time constraints on mean values of the state variable. In order to control the feasibility violation induced by the discretization, error estimates for the semilinear partial differential equation are derived. Based upon these estimates, it can be shown that any local minimizer of the semilinear parabolic optimization problems satisfying a weak second-order sufficient condition can be approximated by the discretized problem. Rates for this convergence in terms of temporal and spatial discretization mesh sizes are provided. In contrast to other results in numerical analysis of optimization problems subject to semilinear parabolic equations, the analysis can work with a weak second-order condition, requiring growth of the Lagrangian in critical directions only. The analysis can then be conducted relying solely on the resulting quadratic growth condition of the continuous problem, without the need for similar assumptions on the discrete or time semidiscrete setting. 相似文献
17.
XQ Yang 《Mathematical Programming》1998,81(3):327-347
In recent years second-order sufficient conditions of an isolated local minimizer for convex composite optimization problems have been established. In this paper, second-order optimality conditions are obtained of aglobal minimizer for convex composite problems with a non-finite valued convex function and a twice strictly differentiable function by introducing a generalized representation condition. This result is applied to a minimization problem with a closed convex set constraint which is shown to satisfy the basic constraint qualification. In particular, second-order necessary and sufficient conditions of a solution for a variational inequality problem with convex composite inequality constraints are obtained. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V. 相似文献
18.
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
19.
In this paper, we present a necessary and sufficient condition for a zero duality gap between a primal optimization problem and its generalized augmented Lagrangian dual problems. The condition is mainly expressed in the form of the lower semicontinuity of a perturbation function at the origin. For a constrained optimization problem, a general equivalence is established for zero duality gap properties defined by a general nonlinear Lagrangian dual problem and a generalized augmented Lagrangian dual problem, respectively. For a constrained optimization problem with both equality and inequality constraints, we prove that first-order and second-order necessary optimality conditions of the augmented Lagrangian problems with a convex quadratic augmenting function converge to that of the original constrained program. For a mathematical program with only equality constraints, we show that the second-order necessary conditions of general augmented Lagrangian problems with a convex augmenting function converge to that of the original constrained program.This research is supported by the Research Grants Council of Hong Kong (PolyU B-Q359.) 相似文献
20.
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