共查询到20条相似文献,搜索用时 15 毫秒
1.
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
2.
《Optimization》2012,61(3):627-640
AbstractIn this work, we give some characterizations of gw-subdifferentiability of a vector-valued function by using its directional derivative and radial epiderivative. Moreover, under some assumptions, we proved that the directional derivative and radial epiderivative of a vector-valued function are the elements of the supremum set of gw-subgradients of it. Finally, without any convexity assumption, we proved that the epigraph of contingent derivative of a set valued map is included in the epigraph of contingent epiderivative of this set-valued map. 相似文献
3.
X. Q. Yang 《Journal of Global Optimization》2004,30(2):271-284
Second-order optimality conditions are studied for the constrained optimization problem where the objective function and the constraints are compositions of convex functions and twice strictly differentiable functions. A second-order sufficient condition of a global minimizer is obtained by introducing a generalized representation condition. Second-order minimizer characterizations for a convex program and a linear fractional program are derived using the generalized representation condition 相似文献
4.
Generalized second-order characterizations of convex functions 总被引:1,自引:0,他引:1
X. Q. Yang 《Journal of Optimization Theory and Applications》1994,82(1):173-180
We introduce a second-order upper Dini-directional derivative and obtain a second-order characterization for a continuously Gâteaux differentiable function to be a convex function.This research was done under the supervision of Dr. V. Jeyakumar whose helpful guidance and valuable suggestions are much appreciated. The author is thankful to the referee for providing a simplication of the original proof of Theorem 2.1. 相似文献
5.
Shunsuke Shiraishi 《Mathematical Programming》1993,58(1-3):257-262
For a real-valued convex functionf, the existence of the second-order Dini derivative assures that of the limit of the approximate second-order directional derivativef
(x
0;d, d) when 0+ and both values are the same. The aim of the present work is to show the converse of this result. It will be shown that upper and lower limits of the approximate second-order directional derivative are equal to the second-order upper and lower Dini derivatives, respectively. Consequently the existence of the limit of the approximate second-order directional derivative and that of second-order Dini derivative are equivalent.Dedicated to Professor N. Furukawa of Kyushu University for his 60th birthday. 相似文献
6.
In this paper, we establish a second-order sufficient condition for constrained optimization problems of a class of so called t-stable functions in terms of the first-order and the second-order Dini type directional derivatives. The result extends the corresponding result of [D. Bednarik and K. Pastor, Math. Program. Ser. A, 113(2008), 283-298] to constrained optimization problems. 相似文献
7.
《Optimization》2012,61(4):305-321
A new general abstract scheme for local second-order approximations and second-order generalized directional derivatives is presented. Applications to optimization are provided. 相似文献
8.
Hidefumi Kawasaki 《Mathematical Programming》1988,41(1-3):73-96
A group of curves generates a new curve which is called an envelope. When one deals with a minimization problem with infinitely many inequality constraints, one must encounter an envelopelike effect caused by the constraints. In this paper we present second-order necessary conditions, which involve a new term besides the second derivative of the Lagrange function.We apply our results to minimizing problems of sup-type functions. One will observe in examples that the new term given in this paper explains well the behavior of the second directional derivative of the sup-type function. 相似文献
9.
The (parabolic) second-order directional derivatives of singular values of matrices and symmetric matrix-valued functions induced by real-valued functions play important roles in studying second-order optimality conditions for different types of matrix cone optimization problems. We propose a direct way to derive the formula for the second-order directional derivative of any eigenvalue of a symmetric matrix in Torki (Nonlinear Anal 46:1133–1150 2001), from which a formula for the second-order directional derivative of any singular value of a matrix is established. We demonstrate a formula for the second-order directional derivative of the symmetric matrix-valued function. As applications, the second-order derivative for the projection operator over the SDP cone is derived and used to get the second-order tangent set of the SDP cone in Bonnans and Shapiro (2000), and the tangent cone and the second-order tangent set of the epigraph of the nuclear norm are given as well. 相似文献
10.
《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
11.
In this paper, a family of parameterized set-valued optimization problems, whose constraint set depends on a parameter, are considered. Some calculus rules are obtained for calculating the second-order contingent derivatives of the composition and sum of two set-valued mappings. Then, by using these calculus rules, some results concerning second-order sensitivity analysis are established, and an explicit expression for the second-order contingent derivative of the (weak) perturbation mapping in the set-valued optimization problems is obtained. 相似文献
12.
D.E. Ward 《Journal of Mathematical Analysis and Applications》2008,348(1):324-336
A chain rule is established for contingent and adjacent epiderivatives and hypoderivatives of compositions g○h, where h is assumed to be Hadamard directionally differentiable. Corollaries include a formula for the contingent and adjacent cones of an equality constraint set defined by a Hadamard directionally differentiable function. An analogous chain rule for second-order contingent and adjacent epiderivatives and hypoderivatives is also developed. 相似文献
13.
Yu Xia 《Computational Optimization and Applications》2007,37(3):371-408
We develop optimality conditions for the second-order cone program. Our optimality conditions are well-defined and smooth
everywhere. We then reformulate the optimality conditions into several systems of equations. Starting from a solution to the
original problem, the sequence generated by Newton’s method converges Q-quadratically to a solution of the perturbed problem
under some assumptions. We globalize the algorithm by (1) extending the gradient descent method for differentiable optimization
to minimizing continuous functions that are almost everywhere differentiable; (2) finding a directional derivative of the
equations. Numerical examples confirm that our algorithm is good for “warm starting” second-order cone programs—in some cases,
the solution of a perturbed instance is hit in two iterations. In the progress of our algorithm development, we also generalize
the nonlinear complementarity function approach for two variables to several variables. 相似文献
14.
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
15.
In this paper, the extremum of second-order directional derivatives, i.e. the gradient of first-order derivatives is discussed.
Given second-order directional derivatives in three nonparallel directions, or given second-order directional derivatives
and mixed directional derivatives in two nonparallel directions, the formulae for the extremum of second-order directional
derivatives are derived, and the directions corresponding to maximum and minimum are perpendicular to each other. 相似文献
16.
17.
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
18.
19.
This paper presents a second-order continuity non-overlapping domain decomposition (DD) technique for numerically solving second-order elliptic problems in two-dimensional space. The proposed DD technique uses integrated Chebyshev polynomials to represent the solution in subdomains. The constants of integration are utilized to impose continuity of the second-order normal derivative of the solution at the interior points of subdomain interfaces. To also achieve a C2 function at the intersection of interfaces, two additional unknowns are introduced at each intersection point. Numerical results show that the present DD method yields a higher level of accuracy than conventional DD techniques based on differentiated Chebyshev polynomials. 相似文献
20.
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