Multiobjective Nonlinear Symmetric Duality Involving Generalized Pseudoconvexity

The purpose of this paper is to introduce second order (K, F)-pseudoconvex and second order strongly (K, F)- pseudoconvex functions which are a generalization of cone-pseudoconvex and strongly cone-pseudoconvex functions. A pair of second order symmetric dual multiobjective nonlinear programs is formulated by using the considered functions. Furthermore, the weak, strong and converse duality theorems for this pair are established. Finally, a self duality theorem is given.

BIG HANKEL OPERATORS ON HARDY SPACES OF STRONGLY PSEUDOCONVEX DOMAINS

In this article,we investigate the(big) Hankel operator H_(f) on the Hardy spaces of bounded strongly pseudoconvex domains Ω in C^(n).We observe that H_(f ) is bounded on H~p(Ω)(1 <p <∞) if f belongs to BMO and we obtain some characterizations for Hf on H^(2)(Ω) of other pseudoconvex domains.In these arguments,Amar's L^(p)-estimations and Berndtsson's L^(2)-estimations for solutions of the ■_(b)-equation play a crucial role.In addition,we solve Gleason's problem for Hardy spaces H^(p)(Ω)(1 ≤p≤∞) of bounded strongly pseudoconvex domains.

Solution Concepts and New Optimality Conditions in Bilevel Multiobjective Programming

In this paper, new sufficient optimality theorems for a solution of a differentiable bilevel multiobjective optimization problem (BMOP) are established. We start with a discussion on solution concepts in bilevel multiobjective programming;a theorem giving necessary and sufficient conditions for a decision vector to be called a solution of the BMOP and a proposition giving the relations between four types of solutions of a BMOP are presented and proved. Then, under the pseudoconvexity assumptions on the upper and lower level objective functions and the quasiconvexity assumptions on the constraints functions, we establish and prove two new sufficient optimality theorems for a solution of a general BMOP with coupled upper level constraints. Two corollary of these theorems, in the case where the upper and lower level objectives and constraints functions are convex are presented.

THE OPTIMALITY CONDITIONS OF NONCONVEXSET-VALUED VECTOR OPTIMIZATION

The concepts of alpha-order Clarke's derivative, alpha-order Adjacent derivative and alpha-order G.Bouligand derivative of set-valued mappings are introduced, their properties are studied, with which the Fritz John optimality condition of set-valued vector optimization is established. Finally, under the assumption of pseudoconvexity, the optimality condition is proved to be sufficient.

A REMARK ON GENERAL COMPLEX(α,β)METRICS

In this paper,we give a characterization for the general complex(α,β)metrics to be strongly convex.As an application,we show that the well-known complex Randers metrics are strongly convex complex Finsler metrics,whereas the complex Kropina metrics are only strongly pseudoconvex.

A NEW FORMULA WITHOUT BOUNDARY INTEGRALS ON A STEIN MANIFOLD

A new Koppelman-Leray-Norguet formula of (p,q) differential forms for a strictly pseudoconvex polyhedron with not necessarily smooth boundary on a Stein manifold is obtained, and an integral representation for the solution of -equation on this domain which does not involve integrals on boundary is given, so one can avoid complex estimates of boundary integrals.

Singular integral on bounded strictly pseudoconvex domain

Kytmanov and Myslivets gave a special Cauchy principal value of the singular integral on the bounded strictly pseudoconvex domain with smooth boundary. By means of this Cauchy integral principal value, the corresponding singular integral and a composition formula are obtained. This composition formula is quite different from usual ones in form. As an application, the corresponding singular integral equation and the system of singular integral equations are discussed as well.

Locally Conformal Pseudo-Kahler Finsler Manifolds

In this paper,we give a necessary and sucient condition for a strongly pseudoconvex complex Finsler metric to be locally conformal pseudo-Kahler Finsler.As an application,we nd any complete strongly convex and locally conformal pseudo-Kahler Finsler manifold,which is simply connected or whose fundamental group contains elements of nite order only,can be given a Kahler metric.

FINITE TYPE CONDITIONS ON REAL HYPERSURFACES WITH ONE DEGENERATE EIGENVALUE

Let M be a smooth pseudoconvex hypersurface in ℂ^(n+1) whose Levi form has at most one degenerate eigenvalue. For any tangent vector field L of type (1, 0), we prove the equality of the commutator type and the Levi form type associated to L. We also show that the regular contact type, the commutator type and the Levi form type of the real hypersurface are the same.

REVISITING A NON-DEGENERACY PROPERTY FOR EXTREMAL MAPPINGS

We extend an earlier result obtained by the author in [7].

_b-EQUATION ON COMPLEX MANIFOLDS

By using Koppelman-Leray kernel and under suitable condition the integral representation formula of equation on complex manifolds is obtained.

L^(2) EXTENSIONS WITH SINGULAR METRICS ON K?HLER MANIFOLDS

In this paper,we give a survey of our recent results on extension theorems on Kähler manifolds for holomorphic sections or cohomology classes of(pluri)canonical line bundles twisted with holomorphic line bundles equipped with singular metrics,and also discuss their applications and the ideas contained in the proofs.

Random gradient-free method for online distributed optimization with strongly pseudoconvex cost functions

This paper focuses on the online distributed optimization problem based on multi-agent systems. In this problem, each agent can only access its own cost function and a convex set, and can only exchange local state information with its current neighbors through a time-varying digraph. In addition, the agents do not have access to the information about the current cost functions until decisions are made. Different from most existing works on online distributed optimization, here we consider the case where the cost functions are strongly pseudoconvex and real gradients of the cost functions are not available. To handle this problem, a random gradient-free online distributed algorithm involving the multi-point gradient estimator is proposed. Of particular interest is that under the proposed algorithm, each agent only uses the estimation information of gradients instead of the real gradient information to make decisions. The dynamic regret is employed to measure the proposed algorithm. We prove that if the cumulative deviation of the minimizer sequence grows within a certain rate, then the expectation of dynamic regret increases sublinearly. Finally, a simulation example is given to corroborate the validity of our results.

Boundary points, minimal L^(2) integrals and concavity property Ⅱ: Weakly pseudoconvex Kähler manifolds

In this paper, we consider minimal L^(2) integrals on the sublevel sets of plurisubharmonic functions on weakly pseudoconvex K?hler manifolds with Lebesgue measurable gain related to modules at boundary points of the sublevel sets, and establish a concavity property of the minimal L^(2) integrals. As applications, we present a necessary condition for the concavity degenerating to linearity, a concavity property related to modules at inner points of the sublevel sets, an optimal support function related to modules, a strong openness property of modules and a twisted version, and an effectiveness result of the strong openness property of modules.

Boundary Points,Minimal L^(2)Integrals and Concavity PropertyⅢ—Linearity on Riemann Surfaces and Fibrations over Open Riemann Surfaces

In this article,we consider a modified version of minimal L^(2) integrals on sublevel sets of plurisubharmonic functions related to modules at boundary points,and obtain a concavity property of the modified version.As applications,we give characterizations for the concavity degenerating to linearity on open Riemann surfaces and on fibrations over open Riemann surfaces.

Zeroth-Order Methods for Online Distributed Optimization with Strongly Pseudoconvex Cost Functions

This paper studies an online distributed optimization problem over multi-agent systems.In this problem,the goal of agents is to cooperatively minimize the sum of locally dynamic cost functions.Different from most existing works on distributed optimization,here we consider the case where the cost function is strongly pseudoconvex and real gradients of objective functions are not available.To handle this problem,an online zeroth-order stochastic optimization algorithm involving the single-point gradient estimator is proposed.Under the algorithm,each agent only has access to the information associated with its own cost function and the estimate of the gradient,and exchange local state information with its immediate neighbors via a time-varying digraph.The performance of the algorithm is measured by the expectation of dynamic regret.Under mild assumptions on graphs,we prove that if the cumulative deviation of minimizer sequence grows within a certain rate,then the expectation of dynamic regret grows sublinearly.Finally,a simulation example is given to illustrate the validity of our results.

Optimality Conditions of Approximate Solutions for Nonsmooth Semi-infinite Programming Problems

In this paper,we study optimality conditions of approximate solutions for nonsmooth semi-infinite programming problems.Three new classes of functions,namelyε-pseudoconvex functions of type I and type II andε-quasiconvex functions are introduced,respectively.By utilizing these new concepts,sufficient optimality conditions of approximate solutions for the nonsmooth semi-infinite programming problem are established.Some examples are also presented.The results obtained in this paper improve the corresponding results of Son et al.(J Optim Theory Appl 141:389–409,2009).

NEUMANN PROBLEM FOR OVERDETERMINED SYSTEM RELATED WITH ALMOST COMPLEX STRUCTURE

In this paper we investigate an overdetermined system of differential equations, which is a generalization of both the Cauchy-Riemann equations and the Beltrami equation. The conditions under which the Neumann problem for the overdetermined system can be solved are given.

COHOMOLOGY VANISHING IN HILBERT SPACES

The authors give two cohomology vanishing theorems for domains, which are not pseudoconvex, and characterize the holomorphy of domains with smooth boundaries in separable Hilbert spaces through cohomology vanishing.

Plurigenera of compact connected strongly pseudoconvex CR manifolds In memory of Salah Baouendi

Strongly pseudoconvex CR manifolds are boundaries of Stein varieties with isolated normal singularities.We introduce a series of new invariant plurigeneraδm,m∈Z+for a strongly pseudoconvex CR manifold.The main purpose of this paper is to present the following result:Let X1and X2be two compact strongly pseudoconvex embeddable CR manifolds of dimension 2n-1 3.If there is a non-constant CR morphism from X1to X2,thenδm(X2)δm(X1)whereδm(Xi)is the plurigeneus of Xi(see Definition 2.4).