On the functions with pseudoconvex sublevel sets and optimality conditions

A new class of generalized convex functions, called the functions with pseudoconvex sublevel sets, is defined. They include quasiconvex ones. A complete characterization of these functions is derived. Further, it is shown that a continuous function admits pseudoconvex sublevel sets if and only if it is quasiconvex. Optimality conditions for a minimum of the nonsmooth nonlinear programming problem with inequality, equality and a set constraints are obtained in terms of the lower Hadamard directional derivative. In particular sufficient conditions for a strict global minimum are given where the functions have pseudoconvex sublevel sets.     

Uniqueness of solutions of aH ∞ optimization problem and complex geometric convexity

We study uniqueness of an H^∞ optimization problem which is central to the worst case frequency domain system design. It was known that if the so-called sublevel sets are strictly convex inC ^N, then the uniqueness holds. On the other hand, there are examples of non-uniqueness if the sublevel sets are just strictly pseudoconvex. In this paper we prove that uniqueness holds for a type of convexity which is strictly in-between geometric and pseudoconvexity.     

THE EXSISTENCE OF $\mu -HOLOMORPHIC$ SEPARATING FUNCTION ON BOUNDED SMOOTH DOMAINS

The Complex analysis of strongly pseudoconvex domains in C^n is rather well known.In this paper it is proved that for a bounded smoothly domain $\omega$ there is a new complex structure on it under which $\omega$ will locally become a strongly convex even though the point on b&\omega& is not a pseudoconvex point from the view of the original complex structure.Particularly if $\omega$ is a weakly pseudoconvex domain,the $\mu$ cam be ,ade siffocoemt;u c;pse tp tje progoma; cp,[;ex strictire.Therefore a lot of properties of strongly pseudoconvex domains will become true on weakly pseudoconvex domains,or general domains.For example,it is proved that there is a $\mu-holomorphic$ separatin function which is holomorphic under the new complex structure.     

On strongly pseudoconvex manifolds with one-dimensional exceptional set

Let X be a strongly pseudoconvex n-dimensional manifold with one-dimensional connected exceptional set S. Here we show that if S is reducible, then X is embeddable into C^N ×P ^{m for some N, m and in particular X is Kählerian with a possible exception for n = 3; we analyze this exceptional case but we do not know if it may occur. The case in which S is irreducible was previously analyzed by Tan [11], who proved that X is embeddable if either n ≠ 3 or S is not isomorphic to} P ¹. In the same paper, Tan gave an example of a non-embeddable and non-Kählerian strongly pseudoconvex three-dimensional manifold withP ¹ as an exceptional set.     

The first eigenvalue of Laplace operator under powers of mean curvature flow

In this paper, we derive the evolution equation for the first eigenvalue of Laplace operator along powers of mean curvature flow. Considering a compact, strictly convex n-dimensional surface M without boundary, which is smoothly immersed in R n+1 , we prove that if the initial 2-dimensional surface M is totally umbilical, then the first eigenvalue is nondecreasing along the unnormalized H k -flow. Moreover, as applications of the evolution equation, we construct some monotonic quantities along this kind of flow.     

Note on generalized convex functions

In this note, an important class of generalized convex functions, called invex functions, is defined under a general framework, and some properties of the functions in this class are derived. It is also shown that a function is (generalized) pseudoconvex if and only if it is quasiconvex and invex.     

On paratopological vector spaces

We show that each first countable paratopological vector space X has a compatible translation invariant quasi-metric such that the open balls are convex whenever X is a pseudoconvex vector space. We introduce the notions of a right-bounded subset and of a right-precompact subset of a paratopological vector space X and prove that X is quasi-normable if and only if the origin has a convex and right-bounded neighborhood. Duality in this context is also discussed. Furthermore, it is shown that the bicompletion of any paratopological vector space (respectively, of any quasi-metric vector space) admits the structure of a paratopological vector space (respectively, of a quasi-metric vector space). Finally, paratopological vector spaces of finite dimension are considered. This revised version was published online in June 2006 with corrections to the Cover Date.     

On the convexifiability of pseudoconvex C2-functions

We present new criteria that characterize functions which are convex transformable by a suitable strictly increasing function. We concentrate on twice continuously differentiable pseudoconvex and strictly pseudoconvex functions, and derive conditions which are both necessary and sufficient for these functions to be convex transformable.     

Local patching for convex sets in Cn

We consider open sets Ω₁ and Ω₂ in ? _n so that $\bar \Omega _1 \cap \bar \Omega _2 = \{ 0\} $ . We investigate the existence of a basis of pseudoconvex neighborhoods for $\bar \Omega _1 \cup \bar \Omega _2 $ and of local (pseudoconvex or polynomially convex) “patching.” Positive results are given for strictly convex sets, and counterexamples are given for strictly pseudoconvex sets (even convex ones).     

On Sets of Measurable Operators Convex and Closed in Topology of Convergence in Measure

For a von Neumann algebra with a faithful normal semifinite trace, the properties of operator “intervals” of three types for operators measurable with respect to the trace are investigated. The first two operator intervals are convex and closed in the topology of convergence in measure, while the third operator interval is convex for all nonnegative operators if and only if the von Neumann algebra is Abelian. A sufficient condition for the operator intervals of the second and third types not to be compact in the topology of convergence in measure is found. For the algebra of all linear bounded operators in a Hilbert space, the operator intervals of the second and third types cannot be compact in the norm topology. A nonnegative operator is compact if and only if its operator interval of the first type is compact in the norm topology. New operator inequalities are proved. Applications to Schatten–von Neumann ideals are obtained. Two examples are considered.     

Characterization of the Hilbert ball by its automorphism group

Let be a bounded, convex domain in a separable Hilbert space. The authors prove a version of the theorem of Bun Wong, which asserts that if such a domain admits an automorphism orbit accumulating at a strongly pseudoconvex boundary point, then it is biholomorphic to the ball. Key ingredients in the proof are a new localization argument using holomorphic peaking functions and the use of new ``normal families' arguments in the construction of the limit biholomorphism.

     

Stable generalization of convex functions

A kind of generalized convex functions is said to be stable with respect to some property (P) if this property is maintaincd during an arbitrary function from this class is disturbed by a linear functional with sufficiently small norm. Unfortunately. known generallzed convexities iike quasicunvexity, explicit quasiconvexity. and pseudoconvexity are not stable with respect to such optimization properties which are expected to be true by these generalizations, even if the domain ol the functions is compact. Therefore, we introduce the notion of s-quasiconvex functions. These functions are quasiconvex, explicitly quasicon vex. and pseudoconvex if they are continuously differentiable. Especially, the s-quasiconvexity is stable with respect to the following important properties: (Pl) all lower level sets are convex, (P2) each local minimum is a global minimum. and (P3) each stationary point is a global minimizer. In this paper, different aspects. of s–quasiconvexity and its stability are investigated.     

Characterizations of pseudoconvex functions and semistrictly quasiconvex ones

In this paper, we provide some new necessary and sufficient conditions for pseudoconvexity and semistrict quasiconvexity of a given proper extended real-valued function in terms of the Clarke–Rockafellar subdifferential. Further, we extend to programs with pseudoconvex objective function two earlier characterizations of the solutions set of a set constrained nonlinear programming problem due to Mangasarian (Oper Res Lett 7:21–26, 1988). A positive function p appears in the most results. It is replaced by the number 1 if the function is convex and its domain of definition is convex, too.     

Canonical Metrics on Generalized Cartan-Hartogs Domains

In this paper, the author considers a class of bounded pseudoconvex domains,i.e., the generalized Cartan-Hartogs domains Ω(μ, m). The first result is that the natural Khler metric g~(Ω(μ,m)) of Ω(μ, m) is extremal if and only if its scalar curvature is a constant. The second result is that the Bergman metric, the Ka¨hler-Einstein metric, the Carathéodary metric, and the Koboyashi metric are equivalent for Ω(μ, m).     

The complex Monge–Ampère equation with infinite boundary value

In this article we consider the complex Monge–Ampère equation with infinite boundary value in bounded pseudoconvex domains. We prove the existence of strictly plurisubharmonic solution to the problem in convex domains under suitable growth conditions. We also obtain, for general pseudoconvex domains, some nonexistence results which show that these growth conditions are nearly optimal.     

Optimality Conditions for Isolated Local Minimum of Nonsmooth Multiobjective Problems

This article is devoted to the study of Fritz John and strong Kuhn-Tucker necessary conditions for properly efficient solutions, efficient solutions and isolated efficient solutions of a nonsmooth multiobjective optimization problem involving inequality and equality constraints and a set constraints in terms of the lower Hadamard directional derivative. Sufficient conditions for the existence of such solutions are also provided where the involved functions have pseudoconvex sublevel sets. Our results are based on the concept of pseudoconvex sublevel sets. The functions with pseudoconvex sublevel sets are a class of generalized convex functions that include quasiconvex functions.     

Remarks on minimal annuli in a slab

We show that the two closed boundary curves of a minimal annulus in a slab are both convex if one of them is convex and along the other curve the surface meets the plane at a constant angle. And therefore, under the same condition, the minimal annulus is foliated by convex planar curves all of which are parallel to the boundary. In particular, if the convex curve is a circle, then the annulus is part of a catenoid.     

Local-global properties of bifunctions

Representations of composite systems, such as bilinear programming, models of consumer/producer behavior, and sensitivity problems involve bifunctions (functions of two vector arguments). Such bifunctions are typically convex, pseudoconvex, or quasiconvex in each of their arguments, but not jointly convex, pseudoconvex, or quasiconvex. These functions do not in general possess the strong local-global property, namely, that every stationary point is a global minimum. In this paper, we define conditions that ensure that a bifunction possesses only a global minimum. In exploring this question, we use P-convexity and pseudo P-convexity, which are classes of bifunctions that generalize quasiconvexity and pseudoconvexity.     

赋范线性空间中的广义凸集

文[1]拓广凸集的概念,在 R~(?) 中引入了伪凸、拟凸等广义凸集的概念,获得了它们的一些性质,因而可使得优化理论的研究更为深入.熟知,逼近理论在优化中的应用是非常广泛的(见[2]),本文试图把广义凸集引入赋范线性空间中,并侧重探究其逼近性质.自然,文[1]在 R~(?) 中得到的广义凸集的一些性质,大多数在赋范空间中都是成立的,且证明