集值优化问题严格有效解的一、二阶最优性条件

首先定义了集值优化问题的m阶局部严格有效解并在赋范空间中研究了解的一些性质.在一定条件下,利用Dini导算子和支撑函数确立了m≥1阶严格有效解存在的充分必要条件.     

Dini derivative and a characterization for Lipschitz and convex functions on Riemannian manifolds

Dini derivatives in Riemannian manifold settings are studied in this paper. In addition, a characterization for Lipschitz and convex functions defined on Riemannian manifolds and sufficient optimality conditions for constraint optimization problems in terms of the Dini derivative are given.     

具有Dini连续性系数的非线性椭圆方程组在自然增长条件下的最优内部部分正则性

本文我们研究的是具有Dini连续性系数的散度形式的非线性椭圆方程组在自然增长条件下的问题．我们证明所用的方法是有Dugaar和Grotowski所引进的调和逼近技巧。这种技窍在证明弱解的局部正则性时非常重要．我们可以用之直接得到最优局部正则性结果．     

椭圆型方程的边界正则性与Dini条件

笔者利用Caffarelli扰动方法,证明了Poisson方程和完全非线性一致椭圆型方程在Dini条件下其粘性解的边界正则性,从而给出了文[8]定理4.11不用位势理论的简化证明.     

Weak minimizers,minimizers and variational inequalities for set-valued functions. A blooming wreath?

Recently, necessary and sufficient conditions in terms of variational inequalities have been introduced to characterize minimizers of convex set-valued functions. Similar results have been proved for a weaker concept of minimizers and weaker variational inequalities. The implications are proved using scalarization techniques that eventually provide original problems, not fully equivalent to the set-valued counterparts. Therefore, we try, in the course of this note, to close the network among the various notions proposed. More specifically, we prove that a minimizer is always a weak minimizer, and a solution to the stronger variational inequality always also a solution to the weak variational inequality of the same type. As a special case, we obtain a complete characterization of efficiency and weak efficiency in vector optimization by set-valued variational inequalities and their scalarizations. Indeed, this might eventually prove the usefulness of the set optimization approach to renew the study of vector optimization.     

Differentiability of solutions to second-order elliptic equations via dynamical systems

For a second-order elliptic equation in divergence form we investigate conditions on the coefficients which imply that all solutions are Lipschitz continuous or differentiable at a given point. We assume the coefficients have modulus of continuity satisfying the square-Dini condition, and obtain additional conditions that examples show are sharp. Our results extend those of previous authors who assume the modulus of continuity satisfies the Dini condition. Our method involves the study of asymptotic properties of solutions to a dynamical system that is derived from the coefficients of the elliptic equation.     

OnG-semidifferentiable functions in Euclidean spaces

We give a criterion for a functionf:R ⁿ R to be upperG-semidifferentiable in the sense of Ref. 1 at a point . Using this result, we describe upperG-semiderivatives whenG is, for instance, one of the following basic classes of homogeneous functions: the set of all continuous positively homogeneous functions, the set of differences of two sublinear functions, and the set of sublinear functions. As a result, connections between upperG-semidifferentiability and the concepts of differentiability in Refs. 2–4 are obtained.This research was supported by a grant from the World Laboratory. The author would like to thank Professor M. Pappalardo for useful comments.     

From Scalar to Vector Optimization

Initially, second-order necessary optimality conditions and sufficient optimality conditions in terms of Hadamard type derivatives for the unconstrained scalar optimization problem ϕ(x) → min, x ∈ ℝ ^m , are given. These conditions work with arbitrary functions ϕ: ℝ ^m → ℝ, but they show inconsistency with the classical derivatives. This is a base to pose the question whether the formulated optimality conditions remain true when the “inconsistent” Hadamard derivatives are replaced with the “consistent” Dini derivatives. It is shown that the answer is affirmative if ϕ is of class (i.e., differentiable with locally Lipschitz derivative). Further, considering functions, the discussion is raised to unconstrained vector optimization problems. Using the so called “oriented distance” from a point to a set, we generalize to an arbitrary ordering cone some second-order necessary conditions and sufficient conditions given by Liu, Neittaanmaki, Krizek for a polyhedral cone. Furthermore, we show that the conditions obtained are sufficient not only for efficiency but also for strict efficiency.     

A note on strong pseudoconvexity

A strongly pseudoconvex function is generalized to non-smooth settings. A complete characterization of the strongly pseudoconvex radially lower semicontinuous functions is obtained.