Duality Theorems for Separable Convex Programming Without Qualifications

In the research of mathematical programming, duality theorems are essential and important elements. Recently, Lagrange duality theorems for separable convex programming have been studied. Tseng proves that there is no duality gap in Lagrange duality for separable convex programming without any qualifications. In other words, although the infimum value of the primal problem equals to the supremum value of the Lagrange dual problem, Lagrange multiplier does not always exist. Jeyakumar and Li prove that Lagrange multiplier always exists without any qualifications for separable sublinear programming. Furthermore, Jeyakumar and Li introduce a necessary and sufficient constraint qualification for Lagrange duality theorem for separable convex programming. However, separable convex constraints do not always satisfy the constraint qualification, that is, Lagrange duality does not always hold for separable convex programming. In this paper, we study duality theorems for separable convex programming without any qualifications. We show that a separable convex inequality system always satisfies the closed cone constraint qualification for quasiconvex programming and investigate a Lagrange-type duality theorem for separable convex programming. In addition, we introduce a duality theorem and a necessary and sufficient optimality condition for a separable convex programming problem, whose constraints do not satisfy the Slater condition.     

Approximate Optimality and Approximate Duality for Quasi Approximate Solutions in Robust Convex Semidefinite Programs

In this paper, we study quasi approximate solutions for a convex semidefinite programming problem in the face of data uncertainty. Using the robust optimization approach (worst-case approach), approximate optimality conditions and approximate duality theorems for quasi approximate solutions in robust convex semidefinite programming problems are explored under the robust characteristic cone constraint qualification. Moreover, some examples are given to illustrate the obtained results.     

Necessary and Sufficient Conditions for Minimax Fractional Programming

We establish the necessary and sufficient optimality conditions for a class of nondifferentiable minimax fractional programming problems solving generalized convex functions. Subsequently, we apply the optimality conditions to formulate one parametric dual problem and we prove weak duality, strong duality, and strict converse duality theorems.     

On optimality and duality theorems of nonlinear disjunctive fractional minmax programs

This paper is concerned with the study of optimality conditions for disjunctive fractional minmax programming problems in which the decision set can be considered as a union of a family of convex sets. Dinkelbach’s global optimization approach for finding the global maximum of the fractional programming problem is discussed. Using the Lagrangian function definition for this type of problem, the Kuhn–Tucker saddle point and stationary-point problems are established. In addition, via the concepts of Mond–Weir type duality and Schaible type duality, a general dual problem is formulated and some weak, strong and converse duality theorems are proven.     

具有(F,α,ρ,d)—凸的分式规划问题的最优性条件和对偶性

给出了一类非线性分式规划问题的参数形式和非参数形式的最优性条件,在此基础上,构造出了一个参数对偶模型和一个非参数对偶模型,并分别证明了其相应的对偶定理,这些结果是建立在次线性函数和广义凸函数的基础上的.     

Complex Minimax Fractional Programming of Analytic Functions

We prove that a minmax fractional programming problem is equivalent to a minimax nonfractional parametric problem for a given parameter in complex space. Using a parametric approach, we establish the Kuhn-Tucker type necessary optimality conditions and prove the existence theorem of optimality for complex minimax fractional programming in the framework of generalized convexity. Subsequently, we apply the optimality conditions to formulate a one-parameter dual problem and prove weak duality, strong duality, and strict converse duality theorems involving generalized convex complex functions. This research was partly supported by NSC, Taiwan.     

Optimality and Duality for Complex Nondifferentiable Fractional Programming

Both parametric and nonparametric necessary and sufficient optimality conditions are established for a class of complex nondifferentiable fractional programming problems containing generalized convex functions. Subsequently, these optimality criteria are utilized as a basis for constructing one parametric and two other parameter-free dual models with appropriate duality theorems.     

一类二层凸规划的对偶二层规划

本文讨论上层目标函数以下层子系统目标函数的最优值作为反馈的一类二层凸规划的对偶规划问题 ,在构成函数满足凸连续可微等条件的假设下 ,建立了二层凸规划的 Lagrange对偶二层规划 ,并证明了基本对偶定理 .     

On sufficiency and duality for a class of interval-valued programming problems

In this paper, we are concerned with an interval-valued programming problem. Sufficient optimality conditions are established under generalized convex functions for a feasible solution to be an efficient solution. Appropriate duality theorems for Mond-Weir and Wolfe type duals are discussed in order to relate the efficient solutions of primal and dual programs.     

Discrete convex analysis

A theory of “discrete convex analysis” is developed for integer-valued functions defined on integer lattice points. The theory parallels the ordinary convex analysis, covering discrete analogues of the fundamental concepts such as conjugacy, subgradients, the Fenchel min-max duality, separation theorems and the Lagrange duality framework for convex/nonconvex optimization. The technical development is based on matroid-theoretic concepts, in particular, submodular functions and exchange axioms. Sections 1–4 extend the conjugacy relationship between submodularity and exchange ability, deepening our understanding of the relationship between convexity and submodularity investigated in the eighties by A. Frank, S. Fujishige, L. Lovász and others. Sections 5 and 6 establish duality theorems for M- and L-convex functions, namely, the Fenchel min-max duality and separation theorems. These are the generalizations of the discrete separation theorem for submodular functions due to A. Frank and the optimality criteria for the submodular flow problem due to M. Iri-N. Tomizawa, S. Fujishige, and A. Frank. A novel Lagrange duality framework is also developed in integer programming. We follow Rockafellar’s conjugate duality approach to convex/nonconvex programs in nonlinear optimization, while technically relying on the fundamental theorems of matroid-theoretic nature.     

On approximate solutions for robust convex semidefinite optimization problems

In this paper, we consider approximate solutions (\(\epsilon \)-solutions) for a convex semidefinite programming problem in the face of data uncertainty. Using robust optimization approach (worst-case approach), we prove an approximate optimality theorem and approximate duality theorems for \(\epsilon \)-solutions in robust convex semidefinite programming problem under the robust characteristic cone constraint qualification. Moreover, an example is given to illustrate the obtained results.     

B-半-（E，F）-凸函数和规划

本文介绍了一组新的广义凸函数: b-半-(E,F)-凸函数(拟b-半-(E,F)-凸函数,伪b-半-(E,F)-凸函数),并讨论了它们的一些基本性质.研究了带不等式约束的非线性b-半-(E,F)-凸规划的最优性充分条件和对偶定理.     

一类Lipschitz B-(p,γ)-不变凸函数与非光滑规划

设本文给出了一类新的Lipschitz B-(p,γ)-不变凸函数,它是B-不变凸函数和(p,γ)-不变凸函数的推广.在这类Lipschitz B-(p,γ)一不变凸性下,建立了非光滑规划的必要和充分最优性条件,讨论了Mond-Weir型对偶和Wolfe型对偶,证明了弱对偶、强对偶和逆对偶定理.所得结果推广了涉及凸函数、B-不变凸函数和(p,γ)-不变凸函数的规划问题的相应结果.     

一般形式多阶段有补偿问题的广义对偶理论

本文在文［１］的基础上，讨论一般形式多阶段有补偿非线性随机规划问题的广义对偶理论与最优化性条件．通过发掘凸规划对偶理论的本质，首先推广了与通常规划问题对偶理论有关的概念的含义，由此构造出所论问题在等价意义下的广义原始泛函与广义对偶泛函，进而得到其广义对偶理论，所得结论不仅能恰当合理地反映问题本身的属性，而且有关定理的表述形式简明、结论较强，可直接应用于多阶段有补偿问题的其它理论研究与数值求解算法的设计中去．上述结果与所用研究方法均推广和发展了通常的对偶理论     

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.     

拓扑向量空间中非光滑向量极值问题的最优性条件与对偶

本文提出了向量值函数的锥D-s凸，锥D-s拟凸，s右导数及锥D-s伪凸等新概念，探讨了锥D-s凸函数的有关性质，建立了带约束非光滑向量极值问题(VP)的最优性必要条件与涉及锥D-s凸(拟凸，伪凸)函数的约束极值问题(VP)的最优性充分条件，给出了原问题(VP)与其Mond-Weir型对偶问题的弱对偶与强对偶结论，揭示了(VP)的局部锥D-(弱)有效解与整体锥D-(弱)有效解，(VP)的锥D-弱有效解与锥D-有效解的关系，所得结果拓广了凸规划及部分广义凸规划的有关结论．     

Lagrange duality in canonical DC programming

We study Lagrange duality theorems for canonical DC programming problems. We show two types Lagrange duality results by using a decomposition method to infinite convex programming problems and by using a previous result by Lemaire (1998)  [6]. Also we observe these constraint qualifications for the duality theorems.     

Sufficiency and Duality in Multiobjective Programming under Generalized Type I Functions

In this paper, new classes of generalized (F,α,ρ,d)-V-type I functions are introduced for differentiable multiobjective programming problems. Based upon these generalized convex functions, sufficient optimality conditions are established. Weak, strong and strict converse duality theorems are also derived for Wolfe and Mond-Weir type multiobjective dual programs.     

多目标半定规划的最优性条件及对偶理论

在不变凸的假设下来讨论多目标半定规划的最优性条件、对偶理论以及非凸半定规划的最优性条件．首先给出了非凸半定规划的一个KKT条件成立的充分必要条件, 并利用此定理证明了其最优性必要条件．其次讨论了多目标半定规划的最优性必要条件、充分条件, 并对其建立Wolfe对偶模型, 证明了弱对偶定理和强对偶定理．     

Optimality and duality in nondifferentiable multiobjective mathematical programming involving higher order (F,α,ρ,d)-type I functions

In this paper, a nondifferentiable multiobjective programming problem is considered where every component of objective and constraint functions contain a term involving the support function of a compact convex set. A new class of higher order (F,α,ρ,d)-type I function is introduced. Necessary optimality conditions and the duality theorems for Wolfe and unified higher order dual problems are established. Several known results can be deduced as special cases.