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

In this paper, we discuss a fairly large number of nonparametric duality results under various generalized (η, ρ)-invexity assumptions for a semi-infinite minmax fractional programming problem.     

Some Properties of Smooth Convex Functions and Newton’s Method

Doklady Mathematics - New properties of convex infinitely differentiable functions related to extremal problems are established. It is shown that, in a neighborhood of the solution, even if the...     

Global Nonparametric Sufficient Optimality Conditions for Semi-Infinite Discrete Minmax Fractional Programming Problems Involving Generalized (η, ρ)-Invex Functions

In this paper, we establish a set of necessary optimality conditions and discuss a fairly large number of sets of global nonparametric sufficient optimality criteria under various generalized (η, ρ)-invexity assumptions for a minmax fractional programming problem with infinitely many nonlinear inequality and equality constraints.     

Second-Order Duality in Multiobjective Programming Involving (F, α, ρ, d)-V-Type I Functions

In this paper, a new class of second-order (F, α, ρ, d)-V-type I functions is introduced that generalizes the notion of (F, α, ρ, θ)-V-convex functions introduced by Zalmai (Computers Math. Appl. 2002; 43:1489–1520) and (F, α, ρ, p, d)-type I functions defined by Hachimi and Aghezzaf (Numer. Funct. Anal. Optim. 2004; 25:725–736). Based on these functions, weak, strong, and strict converse duality theorems are derived for Wolfe and Mond–Weir type multiobjective dual programs in order to relate the efficient and weak efficient solutions of primal and dual problems.     

Higher-Order Duality for Multiobjective Programming Problems Involving (F, α, ρ, d)-V-Type I Functions

A class of functions called higher-order (F, α, ρ, d)-V-type I functions and their generalizations is introduced. Using the assumptions on the functions involved, weak, strong and strict converse duality theorems are established for higher-order Wolfe and Mond-Weir type multiobjective dual programs in order to relate the efficient solutions of primal and dual problems.     

Second Order (F, α, ρ, d,E)-convex function and the Duality Problem

A class of second order (F, α, ρ, d, E)-convex functions and their generalization on functions involved, weak, strong, and converse duality theorems are established for a second order Mond-Weir type dual problem.     

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 : Rn × Rm → Rk and g : Rn × Rm → Rl 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.     

Beyond Convex? Global Optimization is Feasible Only for Convex Objective Functions: A Theorem

It is known that there are feasible algorithms for minimizing convex functions, and that for general functions, global minimization is a difficult (NP-hard) problem. It is reasonable to ask whether there exists a class of functions that is larger than the class of all convex functions for which we can still solve the corresponding minimization problems feasibly. In this paper, we prove, in essence, that no such more general class exists. In other words, we prove that global optimization is always feasible only for convex objective functions.     

On Classes of Generalized Convex Functions, Gordan–Farkas Type Theorems, and Lagrangian Duality

In this paper, we introduce several classes of generalized convex functions already discussed in the literature and show the relation between these classes. Moreover, a Gordan–Farkas type theorem is proved for all these classes and it is shown how these theorems can be used to verify strong Lagrangian duality results in finite-dimensional optimization.     

Semiparametric Sufficient Efficiency Conditions and Duality Models for Multiobjective Fractional Subset Programming Problems with Generalized (Ф, ρ, θ)-Convex Functions

In this paper, we present a multitude of global semiparametric sufficient efficiency and duality results under generalized (, , )-convexity assumptions for a multiobjective fractional subset programming problem.     

Higher-Order (F,α,β,ρ,d,E)-Convexity in Fractional Programming

In this paper we define higher order (F,α,β,ρ,d,E)-convex function with respect to E-differentiable function K and obtain optimality conditions for nonlinear programming problem (NP) from the concept of higher order (F,α,β,ρ,d)-convexity. Here, we establish Mond-Weir and Wolfe duality for (NP) and utilize these duality in nonlinear fractional programming problem.     

一类G-(F,ρ)凸多目标分式规划的最优性条件

本文讨论了一类多目标分式规划问题，其中所包含的函数是局部Lipschitz的和Clarke次可微的．首先，在G-(F，ρ)凸的条件下，证明了择一定理．然后，证明了该多目标分式规划问题在Geoffrion意义下的真有效解的充分条件和必要条件．     

关于ρ(L_(r,w))、ρ(J_w)的最小值

本文证明了当线性方程组系数矩阵 A之 Jacobi迭代矩阵 B=L+ U≥ 0 ,ρ( B) <1时 Gauss-Seidel法之迭代矩阵 G=L1,1的谱半径 ρ( G) =ρ( L1,1)是 ρ( Lr,w) ( 0≤ r≤w≤ 1 ,w>0 )中的最小值 ,即此时 Gauss-Seidel迭代是 AOR法中收敛最快的迭代法 .并且对 JOR法 (谱半径为 ρ( Jw) )和 SAOR法也作了相应的论述 .     

On σ(·, W)-Holomorphic Functions and Theorems of Vitali-Type

The aim of this paper is to give some criterions for holomorphy of F-valued σ(F, W)-holomorphic functions which are bounded on bounded sets in a domain D of Fréchet spaces E (resp. ${\mathbb{C}^n}$ ) where ${W \subset F'}$ defines the topology of Fréchet space F. Base on these results we consider the problem on holomorphic extension of F-valued σ(F, W)-holomorphic functions from non-rare subsets of D and from subsets of D which determines uniform convergence in H(D). As an application of the above, some theorems of Vitali-type for a locally bounded sequence ${\{f_i\}_{i \in \mathbb{N}}}$ of Fréchet-valued holomorphic functions are also proved.