Duality theorem for smash coproduct over quantum groupoids

The duality theorem of generalized weak smash coproducts of weak module coalgebras and comodule coalgebras over quantum groupoids is studied.Let H be a weak Hopf algebra,C a left weak H-comodule coalgebra and D a left weak H-module coalgebra.First,a weak generalized smash coproduct C×lH D over quantum groupoids is defined and the module and comodule structures on it are constructed.The weak generalized right smash coproduct C×rL D is similar.Then some isomorph-isms between them are obtained.Secondly,by introducing some concepts of a weak convolution invertible element,a weak co-inner coaction and a strongly relative co-inner coaction,a sufficient condition for C×rH D to be isomorphic to Cv D is obtained,where v∈WC（C,H）and the coaction of H on D is right strongly relative co-inner.Finally,the duality theorem for a generalized smash coproduct over quantum groupoids,（C×lH H）×lH H≌Cv（H×lH H）,is obtained.

Duality theorem for weak L-R smash products

This paper gives a duality theorem for weak L-R smash products, which extends the duality theorem for weak smash products given by Nikshych.

Optimality conditions and duality for a class of nondifferentiable multiobjective generalized fractional programming problems

This paper studies a class of multiobjective generalized fractional programming problems, where the numerators of objective functions are the sum of differentiable function and convex function, while the denominators are the difference of differentiable function and convex function. Under the assumption of Calmness Constraint Qualification the Kuhn-Tucker type necessary conditions for efficient solution are given, and the Kuhn-Tucker type sufficient conditions for efficient solution are presented under the assumptions of （F, α, ρ, d）-V-convexity. Subsequently, the optimality conditions for two kinds of duality models are formulated and duality theorems are proved.

Some Duality Theorems in Nonsmooth Generalized Convex Programming

In this paper, we use the concepts of (F, ρ) invexity, (F, ρ) quasiinvexity and (F,ρ) pseudoinvexity etc. to discuss the generalized convex programming min f(x) s.t. g i(x) 0 (i=1,…,m) g j(x)=0 (j=1,…, n), x ∈ Xand obtain various duality theorems. Where X is a Frechet space. f,g i(i=1,…, m) and h j(j=1, …, n) are functions from X to R 1.

On Nondifferentiable Higher-Order Symmetric Duality in Multiobjective Programming Involving Cones

In this paper,we point out some deficiencies in a recent paper(Lee and Kim in J.Nonlinear Convex Anal.13:599–614,2012),and we establish strong duality and converse duality theorems for two types of nondifferentiable higher-order symmetric duals multiobjective programming involving cones.

Parametric Duality Models for Semi-infinite Discrete Minmax Fractional Programming Problems Involving Generalized (η,ρ)-Invex Functions

A semi-infinite programming problem is a mathematical programming problem with a finite number of variables and infinitely many constraints. Duality theories and generalized convexity concepts are important research topics in mathematical programming. In this paper, we discuss a fairly large number of paramet- ric duality results under various generalized （η,ρ）-invexity assumptions for a semi-infinite minmax fractional programming problem.

Generalized Multiobjective Symmetric Duality under Second-Order(F, α, ρ, d)-Convexity

In this paper, we first formulate a second-order multiobjective symmetric primal-dual pair over arbitrary cones by introducing two different functions f ： R^n × R^m → Rk and g ： R^n × R^m → R^l in each k-objectives as well as l-constraints. Further, appropriate duality relations are established under second-order（F, α, ρ, d）-convexity assumptions. A nontrivial example which is second-order（F, α, ρ, d）-convex but not secondorder convex/F-convex is also illustrated. Moreover, a second-order minimax mixed integer dual programs is formulated and a duality theorem is established using second-order（F, α, ρ, d）-convexity assumptions. A self duality theorem is also obtained by assuming the functions involved to be skew-symmetric.

Parametric Duality Models for Semiinfinite Multiobjective Fractional Programming Problems Containing Generalized (α, η, ρ)-V-Invex Functions

In this paper, we present several parametric duality results under various generalized （a,v,p）-V- invexity assumptions for a semiinfinite multiobjective fractional programming problem.

DUALITY FOR A CLASS OF NONS MOOTH MULTIOBJECTIVE PROGRAMMING PROBLEMS

In this paper, two new dual models of nonsmooth multiobjective programmingare constructed and two duality results are derived.

A New Approach on Mixed-Type Nondifferentiable Higher-Order Symmetric Duality

In this paper,a new mixed-type higher-order symmetric duality in scalar-objective programming is formulated.In the literature we have results either Wolfe or Mond–Weir-type dual or separately,while in this we have combined those results over one model.The weak,strong and converse duality theorems are proved for these programs underη-invexity/η-pseudoinvexity assumptions.Self-duality is also discussed.Our results generalize some existing dual formulations which were discussed by Agarwal et al.(Generalized second-order mixed symmetric duality in nondifferentiable mathematical programming.Abstr.Appl.Anal.2011.https://doi.org/10.1155/2011/103597),Chen(Higher-order symmetric duality in nonlinear nondifferentiable programs),Gulati and Gupta(Wolfe type second order symmetric duality in nondifferentiable programming.J.Math.Anal.Appl.310,247–253,2005,Higher order nondifferentiable symmetric duality with generalized F-convexity.J.Math.Anal.Appl.329,229–237,2007),Gulati and Verma(Nondifferentiable higher order symmetric duality under invexity/generalized invexity.Filomat 28(8),1661–1674,2014),Hou andYang(On second-order symmetric duality in nondifferentiable programming.J Math Anal Appl.255,488–491,2001),Verma and Gulati(Higher order symmetric duality using generalized invexity.In:Proceeding of 3rd International Conference on Operations Research and Statistics(ORS).2013.https://doi.org/10.5176/2251-1938_ORS13.16,Wolfe type higher order symmetric duality under invexity.J Appl Math Inform.32,153–159,2014).

Higher-order Mond-Weir converse duality in multiobjective programming involving cones

In this work, we established a converse duality theorem for higher-order Mond-Weir type multiob- jective programming involving cones. This fills some gap in recently work of Kim et al. [Kim D S, Kang H S, Lee Y J, et al. Higher order duality in inultiobjective programming with cone constraints. Optimization, 2010, 59： 29-43].

OPTIMIZATION AND DUALITY OF CONE-D.C.PROGRAMMING

This paper gives the definitions and some properties of E-directional derivate and ε-subgradientsof cone-convex function. mom them, the optimality conditions of local and global optimal point of unconstrained cone-d.c. programming are gained. At last, the duality theorems of this programming are presented.

A Note of Duality Theorem for Decomposable Operators

St. Frunza"3 laid the foundation of-duality theorem for decomposable operators. He proved that if y￡J?(X) is a 2-decomposable operator, then T* is also a 2-decomposable operator. His proof is fundamental, but it is rather complicated. Here we give a very simple proof, using Dieudonn^’s duality theorem.Wang Shengwang proved that if T* is a decomposable operator, and F a closed set, then -STJ.^) is weak* closed. In this paper we prove if T* has the single-valued extension property, then xt*(f~) is weak* closed whenever it is closed in the strong topology.

Spacelike Willmore surfaces in 4-dimensional Lorentzian space forms

Spacelike Willmore surfaces in 4-dimensional Lorentzian space forms, a topic in Lorentzian conformal geometry which parallels the theory of Willmore surfaces in S4, are studied in this paper. We define two kinds of transforms for such a surface, which produce the so-called left/right polar surfaces and the adjoint surfaces. These new surfaces are again conformal Willmore surfaces. For them the interesting duality theorem holds. As an application spacelike Willmore 2-spheres are classified. Finally we construct a family of homogeneous spacelike Willmore tori.

Portfolio Selection with Random Liability and Affine Interest Rate in the Mean-Variance Framework

This paper studies a dynamic mean-variance portfolio selection problem with random liability in the affine interest rate environment, where the financial market consists of three assets: one risk-free asset, one risky asset and one zero-coupon bond. Assume that short rate is driven by affine interest rate model and liability process is described by the drifted Brownian motion, in addition, stock price dynamics is affected by interest rate dynamics. The investors expect to look for an optimal strategy to minimize the variance of the terminal surplus for a given expected terminal surplus. The efficient strategy and the efficient frontier are explicitly obtained by applying dynamic programming principle and Lagrange duality theorem. A numerical example is given to illustrate our results and some economic implications are analyzed.