Second-Order Modified Contingent Epiderivatives in Set-Valued Optimization

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.     

Some theoretical limitations of second-order algorithms for smooth constrained optimization

In second-order algorithms, we investigate the relevance of the constant rank of the full set of active constraints in ensuring global convergence to a second-order stationary point under a constraint qualification. We show that second-order stationarity is not expected in the non-constant rank case if the growth of so-called tangent AKKT2 sequences is not controlled. Since no algorithm controls their growth, we argue that there is a theoretical limitation of algorithms in finding second-order stationary points without constant rank assumptions.     

Second-order global optimality conditions for convex composite optimization

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.     

二阶微分方程的逆特征值问题

运用渐近分析的方法及Rayleigh商原理,将Sturm-Liouville问题的Ambarzumyan定理推广到具有Neumann边界条件或拟周期边界条件的二阶微分方程情形.同时,获得了二阶向量微分方程的有关Ambarzumyan型结果.     

二阶微分方程的逆特征值问题

运用渐近分析的方法及Rayleigh商原理,将Sturm-Liouville问题的Ambarzumyan定理推广到具有Neumann边界条件或拟周期边界条件的二阶微分方程情形.同时,获得了二阶向量微分方程的有关Ambarzumyan型结果.     

First- and second-order directional differentiability of locally Lipschitzian functions

In this paper, characterizations of the existence of the directional derivative and second-order parabolic directional derivative of a locally Lipschitzian function are established. These characterizations involve the adjacent cone and second-order adjacent set of the graph of the function.     

New Results on Second-Order Optimality Conditions in Vector Optimization Problems

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.     

锥约束多目标规划的二阶对称对偶性质

给出一对锥约束多目标非线性规划的二阶对称对偶问题,以及二阶F凸函数类的概念.在二阶F凸假设下证明了真有效解的对偶性质———弱对偶性、强对偶性及逆对偶性.     

A second-order continuity domain decomposition technique based on integrated Chebyshev polynomials for two-dimensional elliptic problems

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$C^2$ 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.     

Optimality Conditions in Differentiable Vector Optimization via Second-Order Tangent Sets

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.     

Asymptotic behavior of solutions of second-order nonlinear impulsive differential equations

In this work, the asymptotic behavior of all solutions of second-order nonlinear ordinary differential equations with impulses is investigated. By impulsive differential inequality and Riccati transformation, sufficient conditions of asymptotic behavior of all solutions of second-order nonlinear ordinary differential equations with impulses are obtained. An example is also inserted to illustrate the impulsive effect.     

A symmetry completeness criterion for second-order differential equations

A simple criterion for the completeness of an infinitesimal automorphism of a second-order differential equation is given.

     

On the estimation of the shape parameter of the gamma distribution in second-order asymptotics

In the two-parameter case with parameter orthogonality, we propose a method of constructing an estimator with second-order admissibility under any loss function with a given loss coefficient. Furthermore, we give a sufficient condition for any estimator to be second-order admissible or inadmissible. On the basis of these results, the problem of estimating the shape parameter of the gamma distribution is discussed at the level of second-order asymptotics.     

Oscillation and Nonoscillation Criteria for Second-Order Linear Differential Equations

Sufficient conditions for oscillation and nonoscillation of second-order linear equations are established.     

Parabolic equations with measurable coefficients II

We prove the existence and uniqueness of solutions in Sobolev spaces to second-order parabolic equations in non-divergence form. The coefficients (except one of them) of second-order terms of the equations are measurable in both time and one spatial variables, and VMO (vanishing mean oscillation) in other spatial variables.     

Properties of saddle points for generalized augmented Lagrangian

For inequality constrained optimization problem, we show the existence of local saddle point of generalized augmented Lagrangian under weak second-order sufficient conditions which are weaker than the second-order sufficient conditions in the literature. We further discuss the existence of global saddle points without requiring the uniqueness of the global optimal solution. This work was supported by the National Natural Science Foundation of China grants 10571106, 10471159.     

Generalized second-order characterizations of convex functions

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.     

On constraint qualifications for second-order optimality conditions depending on a single Lagrange multiplier

We present new constraint qualifications (CQs) to ensure the validity of some well-known second-order optimality conditions. Our main interest is on second-order conditions that can be associated with numerical methods for solving constrained optimization problems. Such conditions depend on a single Lagrange multiplier, instead of the whole set of Lagrange multipliers. For each condition, we characterize the weakest CQ that guarantees its fulfillment at local minimizers, while proposing new weak conditions implying them. Relations with other CQs are discussed.     

Non-standard Analysis in WKL0

Within a weak subsystem of second-order arithmetic WKL₀, we develop basic part of non-standard analysis up to the Peano existence theorem.