A modified objective function method for solving nonlinear multiobjective fractional programming problems

A new method is used for solving nonlinear multiobjective fractional programming problems having V-invex objective and constraint functions with respect to the same function η. In this approach, an equivalent vector programming problem is constructed by a modification of the objective fractional function in the original nonlinear multiobjective fractional problem. Furthermore, a modified Lagrange function is introduced for a constructed vector optimization problem. By the help of the modified Lagrange function, saddle point results are presented for the original nonlinear fractional programming problem with several ratios. Finally, a Mond-Weir type dual is associated, and weak, strong and converse duality results are established by using the introduced method with a modified function. To obtain these duality results between the original multiobjective fractional programming problem and its original Mond-Weir duals, a modified Mond-Weir vector dual problem with a modified objective function is constructed.     

Multiobjective duality for convex ratios

In this paper we present a duality approach for a multiobjective fractional programming problem. The components of the vector objective function are particular ratios involving the square of a convex function and a positive concave function. Applying the Fenchel-Rockafellar duality theory for a scalar optimization problem associated to the multiobjective primal, a dual problem is derived. This scalar dual problem is formulated in terms of conjugate functions and its structure gives an idea about how to construct a multiobjective dual problem in a natural way. Weak and strong duality assertions are presented.     

有界整规划中的渐近强非线性对偶

本文提出了一种整数规划中的指数一对数对偶.证明了此指数-对数对偶方法具有的渐近强对偶性质,并提出了不需要进行对偶搜索来解原整数规划问题的方法.特别地,当选取合适的参数和对偶变量时,原整数规划问题的解可以通过解一个非线性松弛问题来得到.对具有整系数目标函数及约束函数的多项式整规划问题,给出了参数及对偶变量的取法.     

Saddle points and gap functions for vector equilibrium problems via conjugate duality in vector optimization

In this paper, two conjugate dual problems based on weak efficiency to a constrained vector optimization problem are introduced. Some inclusion relations between the dual objective mappings and the properties of the Lagrangian maps and their saddle points for primal problem are discussed. Gap functions for a vector equilibrium problem are established by using the weak and strong duality.     

Gap functions for generalized vector equilibrium problems via conjugate duality and applications

In this article, gap functions for a generalized vector equilibrium problem (GVEP) with explicit constraints are investigated. Under a concept of supremum/infimum of a set, defined in terms of a closure of the set, three kinds of conjugate dual problems are investigated by considering the different perturbations to GVEP. Then, gap functions for GVEP are established by using the weak and strong duality results. As application, the proposed approach is applied to construct gap functions for a vector optimization problem and a generalized vector variational inequality problem.     

Conjugate maps and duality in multiobjective optimization

This paper considers duality in convex vector optimization. A vector optimization problem requires one to find all the efficient points of the attainable value set for given multiple objective functions. Embedding the primal problem into a family of perturbed problems enables one to define a dual problem in terms of the conjugate map of the perturbed objective function. Every solution of the stable primal problem is associated with a certain solution of the dual problem, which is characterized as a subgradient of the perturbed efficient value map. This pair of solutions also provides a saddle point of the Lagrangian map.     

Classical linear vector optimization duality revisited

With this note we bring again into attention a vector dual problem neglected by the contributions who have recently announced the successful healing of the trouble encountered by the classical duals to the classical linear vector optimization problem. This vector dual problem has, different to the mentioned works which are of set-valued nature, a vector objective function. Weak, strong and converse duality for this “new-old” vector dual problem are proven and we also investigate its connections to other vector duals considered in the same framework in the literature. We also show that the efficient solutions of the classical linear vector optimization problem coincide with its properly efficient solutions (in any sense) when the image space is partially ordered by a nontrivial pointed closed convex cone, too.     

An extension of gap functions for a system of vector equilibrium problems with applications to optimization problems

In this paper, the notion of gap functions is extended from scalar case to vector one. Then, gap functions and generalized functions for several kinds of vector equilibrium problems are shown. As an application, the dual problem of a class of optimization problems with a system of vector equilibrium constraints (in short, OP) is established, the concavity of the dual function, the weak duality of (OP) and the saddle point sufficient condition are derived by using generalized gap functions. This work was supported by the National Natural Science Foundation of China (10671135) and the Applied Research Project of Sichuan Province (05JY029-009-1).     

A method of duality for a mixed vector equilibrium problem

In this paper, a dual scheme for a mixed vector equilibrium problem is introduced by using the method of Fenchel conjugate function. Under the stabilization condition, the relationships between the solutions of mixed vector equilibrium problem (MVEP) and dual mixed vector equilibrium problem (DMVEP) are discussed. Moreover, under the same condition, the solutions of MVEP and DMVEP are proved relating to the saddle points of an associated Lagrangian mapping. As applications, this dual scheme is applied to vector convex optimization and vector variational inequality.     

Employing different loss functions for the classification of images via supervised learning

Supervised learning methods are powerful techniques to learn a function from a given set of labeled data, the so-called training data. In this paper the support vector machines approach is applied to an image classification task. Starting with the corresponding Tikhonov regularization problem, reformulated as a convex optimization problem, we introduce a conjugate dual problem to it and prove that, whenever strong duality holds, the function to be learned can be expressed via the dual optimal solutions. Corresponding dual problems are then derived for different loss functions. The theoretical results are applied by numerically solving a classification task using high dimensional real-world data in order to obtain optimal classifiers. The results demonstrate the excellent performance of support vector classification for this particular problem.     

Higher order duality for a new class of nonconvex semi-infinite multiobjective fractional programming with support functions

In the paper, a new class of semi-infinite multiobjective fractional programming problems with support functions in the objective and constraint functions is considered. For such vector optimization problems, higher order dual problems in the sense of Mond-Weir and Schaible are defined. Then, various duality results between the considered multiobjective fractional semi-infinite programming problem and its higher order dual problems mentioned above are established under assumptions that the involved functions are higher order $\left(\Phi,\rho,\sigma^{\alpha}\right)$-type I functions. The results established in the paper generalize several similar results previously established in the literature.     

Optimality conditions for weak efficiency to vector optimization problems with composed convex functions

We consider a convex optimization problem with a vector valued function as objective function and convex cone inequality constraints. We suppose that each entry of the objective function is the composition of some convex functions. Our aim is to provide necessary and sufficient conditions for the weakly efficient solutions of this vector problem. Moreover, a multiobjective dual treatment is given and weak and strong duality assertions are proved.      

Characterization of vector strict global minimizers of order 2 in differentiable vector optimization problems under a new approximation method

In this paper, a new approximation method is introduced to characterize a so-called vector strict global minimizer of order 2 for a class of nonlinear differentiable multiobjective programming problems with (F,ρ)-convex functions of order 2. In this method, an equivalent vector optimization problem is constructed by a modification of both the objectives and the constraint functions in the original multiobjective programming problem at the given feasible point. In order to prove the equivalence between the original multiobjective programming problem and its associated F-approximated vector optimization problem, the suitable (F,ρ)-convexity of order 2 assumption is imposed on the functions constituting the considered vector optimization problem.     

Extended Farkas lemma and strong duality for composite optimization problems with DC functions

In this paper, we consider the optimization problem in locally convex Hausdorff topological vector spaces with objectives given as the difference of two composite functions and constraints described by an arbitrary (possibly infinite) number of convex inequalities. Using the epigraph technique, we introduce some new constraint qualifications, which completely characterize the Farkas lemma, the dualities between the primal problem and its dual problem. Applications to the conical programming with DC composite function are also given.     

向量优化中广义增广拉格朗日对偶理论及应用

作者介绍了一种基于向量值延拓函数的广义增广拉格朗日函数,建立了基于广义增广拉格朗日函数的集值广义增广拉格朗日对偶映射和相应的对偶问题,得到了相应的强对偶和弱对偶结果,将所获结果应用到约束向量优化问题.该文的结果推广了一些已有的结论.     

Revisiting the Concentration Problem of Vector Fields within a Spherical Cap: A Commuting Differential Operator Solution

We propose a novel basis of vector functions, the mixed vector spherical harmonics that are closely related to the functions of Sheppard and Török and help us reduce the concentration problem of tangential vector fields within a spherical cap to an equivalent scalar problem. Exploiting an analogy with previous results published by Grünbaum, Longhi and Perlstadt, we construct a differential operator that commutes with the concentration operator of this scalar problem and propose a stable and convenient method to obtain its eigenfunctions. Having obtained the scalar eigenfunctions, the calculation of tangential vector Slepian functions is straightforward.     

Optimally Conditions for Pareto Nonsmooth Nonconvex Programming in Banach Spaces

In this paper, we consider a vector optimization problem where all functions involved are defined on Banach spaces. We obtain necessary and sufficient criteria for optimality in the form of Karush–Kuhn–Tucker conditions. We also introduce a nonsmooth dual problem and provide duality theorems.     

Duality Theory for Optimization Problems with Interval-Valued Objective Functions

A solution concept in optimization problems with interval-valued objective functions, which is essentially similar to the concept of nondominated solution in vector optimization problems, is introduced by imposing a partial ordering on the set of all closed intervals. The interval-valued Lagrangian function and interval-valued Lagrangian dual function are also proposed to formulate the dual problem of the interval-valued optimization problem. Under this setting, weak and strong duality theorems can be obtained.     

An η-approximation approach to duality in mathematical programming problems involving r-invex functions

An η-approximation approach introduced by Antczak [T. Antczak, A new method of solving nonlinear mathematical programming problems involving r-invex functions, J. Math. Anal. Appl. 311 (2005) 313-323] is used to obtain a solution Mond-Weir dual problems involving r-invex functions. η-Approximated Mond-Weir dual problems are introduced for the η-approximated optimization problem constructed in this method associated with the original nonlinear mathematical programming problem. By the help of η-approximated dual problems various duality results are established for the original mathematical programming problem and its original Mond-Weir duals.