Comparison between different duals in multiobjective fractional programming

The present paper is a continuation of [2] where we deal with the duality for a multiobjective fractional optimization problem. The basic idea in [2] consists in attaching an intermediate multiobjective convex optimization problem to the primal fractional problem, using an approach due to Dinkelbach ([6]), for which we construct then a dual problem expressed in terms of the conjugates of the functions involved. The weak, strong and converse duality statements for the intermediate problems allow us to give dual characterizations for the efficient solutions of the initial fractional problem. The aim of this paper is to compare the intermediate dual problem with other similar dual problems known from the literature. We completely establish the inclusion relations between the image sets of the duals as well as between the sets of maximal elements of the image sets.      

A new Fenchel dual problem in vector optimization

We introduce a new Fenchel dual for vector optimization problems inspired by the form of the Fenchel dual attached to the scalarized primal multiobjective problem. For the vector primal-dual pair we prove weak and strong duality. Furthermore, we recall two other Fenchel-type dual problems introduced in the past in the literature, in the vector case, and make a comparison among all three duals. Moreover, we show that their sets of maximal elements are equal.     

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.     

Duality for multiobjective optimization problems with convex objective functions and D.C. constraints

In this paper we provide a duality theory for multiobjective optimization problems with convex objective functions and finitely many D.C. constraints. In order to do this, we study first the duality for a scalar convex optimization problem with inequality constraints defined by extended real-valued convex functions. For a family of multiobjective problems associated to the initial one we determine then, by means of the scalar duality results, their multiobjective dual problems. Finally, we consider as a special case the duality for the convex multiobjective optimization problem with convex constraints.     

多目标分式变分问题的混合对偶性

在广义B-Ⅰ凸性条件下,建立了多目标分式变分问题的混合对偶模型,使得M ond-W e ir型对偶和W o lfe型成为其特殊情况,并建立了关于有效解的混合对偶理论.     

On the relations between different duals assigned to composed optimization problems

For an optimization problem with a composed objective function and composed constraint functions we determine, by means of the conjugacy approach based on the perturbation theory, some dual problems to it. The relations between the optimal objective values of these duals are studied. Moreover, sufficient conditions are given in order to achieve equality between the optimal objective values of the duals and strong duality between the primal and the dual problems, respectively. Finally, some special cases of this problem are presented.      

Duality for multiobjective control problems with generalized invexity

Wolfe and Mond-Weir type duals for multiobjective control problems are formulated. Under pseudo-invexity/quasi-invexity assumptions on the functions involved, weak and strong duality theorems are proved to relate efficient solutions of the primal and dual problems.     

Robust conjugate duality for convex optimization under uncertainty with application to data classification

In this paper we present a robust conjugate duality theory for convex programming problems in the face of data uncertainty within the framework of robust optimization, extending the powerful conjugate duality technique. We first establish robust strong duality between an uncertain primal parameterized convex programming model problem and its uncertain conjugate dual by proving strong duality between the deterministic robust counterpart of the primal model and the optimistic counterpart of its dual problem under a regularity condition. This regularity condition is not only sufficient for robust duality but also necessary for it whenever robust duality holds for every linear perturbation of the objective function of the primal model problem. More importantly, we show that robust strong duality always holds for partially finite convex programming problems under scenario data uncertainty and that the optimistic counterpart of the dual is a tractable finite dimensional problem. As an application, we also derive a robust conjugate duality theorem for support vector machines which are a class of important convex optimization models for classifying two labelled data sets. The support vector machine has emerged as a powerful modelling tool for machine learning problems of data classification that arise in many areas of application in information and computer sciences.     

Duality theorems for a new class of multitime multiobjective variational problems

In this work, we consider a new class of multitime multiobjective variational problems of minimizing a vector of functionals of curvilinear integral type. Based on the normal efficiency conditions for multitime multiobjective variational problems, we study duals of Mond-Weir type, generalized Mond-Weir-Zalmai type and under some assumptions of (??, b)-quasiinvexity, duality theorems are stated. We give weak duality theorems, proving that the value of the objective function of the primal cannot exceed the value of the dual. Moreover, we study the connection between values of the objective functions of the primal and dual programs, in direct and converse duality theorems. While the results in §1 and §2 are introductory in nature, to the best of our knowledge, the results in §3 are new and they have not been reported in literature.     

Strong Duality with Proper Efficiency in Multiobjective Optimization Involving Nonconvex Set-Valued Maps

In this paper, we consider some dual problems of a primal multiobjective problem involving nonconvex set-valued maps. For each dual problem, we give conditions under which strong duality between the primal and dual problems holds in the sense that, starting from a Benson properly efficient solution of the primal problem, we can construct a Benson properly efficient solution of the dual problem such that the corresponding objective values of both problems are equal. The notion of generalized convexity of set-valued maps we use in this paper is that of near-subconvexlikeness.     

多目标分式规划逆对偶研究

考虑了一类可微多目标分式规划问题.首先,建立原问题的两个对偶模型.随后,在相关文献的弱对偶定理基础上,利用Fritz John型必要条件,证明了相应的逆对偶定理.     

Robust duality for generalized convex programming problems under data uncertainty

In this paper we present a robust duality theory for generalized convex programming problems in the face of data uncertainty within the framework of robust optimization. We establish robust strong duality for an uncertain nonlinear programming primal problem and its uncertain Lagrangian dual by showing strong duality between the deterministic counterparts: robust counterpart of the primal model and the optimistic counterpart of its dual problem. A robust strong duality theorem is given whenever the Lagrangian function is convex. We provide classes of uncertain non-convex programming problems for which robust strong duality holds under a constraint qualification. In particular, we show that robust strong duality is guaranteed for non-convex quadratic programming problems with a single quadratic constraint with the spectral norm uncertainty under a generalized Slater condition. Numerical examples are given to illustrate the nature of robust duality for uncertain nonlinear programming problems. We further show that robust duality continues to hold under a weakened convexity condition.     

Lagrangian function and duality theory in multiobjective programming with set functions

Using the concept of vector-valued Lagrangian functions, we characterize a special class of solutions,D-solutions, of a multiobjective programming problem with set functions in which the domination structure is described by a closed convex coneD. Properties of two perturbation functions, primal map and dual map, are also studied. Results lead to a general duality theorem.The authors greatly appreciate helpful and valuable comments and suggestions received from the referee.     

On <Emphasis Type="Italic">G</Emphasis>-invex multiobjective programming. Part II. Duality

This paper represents the second part of a study concerning the so-called G-multiobjective programming. A new approach to duality in differentiable vector optimization problems is presented. The techniques used are based on the results established in the paper: On G-invex multiobjective programming. Part I. Optimality by T.Antczak. In this work, we use a generalization of convexity, namely G-invexity, to prove new duality results for nonlinear differentiable multiobjective programming problems. For such vector optimization problems, a number of new vector duality problems is introduced. The so-called G-Mond–Weir, G-Wolfe and G-mixed dual vector problems to the primal one are defined. Furthermore, various so-called G-duality theorems are proved between the considered differentiable multiobjective programming problem and its nonconvex vector G-dual problems. Some previous duality results for differentiable multiobjective programming problems turn out to be special cases of the results described in the paper.     

Duality theory in fuzzy optimization problems formulated by the Wolfe’s primal and dual pair

The weak and strong duality theorems in fuzzy optimization problem based on the formulation of Wolfe’s primal and dual pair problems are derived in this paper. The solution concepts of primal and dual problems are inspired by the nondominated solution concept employed in multiobjective programming problems, since the ordering among the fuzzy numbers introduced in this paper is a partial ordering. In order to consider the differentiation of a fuzzy-valued function, we invoke the Hausdorff metric to define the distance between two fuzzy numbers and the Hukuhara difference to define the difference of two fuzzy numbers. Under these settings, the Wolfe’s dual problem can be formulated by considering the gradients of differentiable fuzzy- valued functions. The concept of having no duality gap in weak and strong sense are also introduced, and the strong duality theorems in weak and strong sense are then derived naturally.     

Dynamic convex duality in constrained utility maximization

ABSTRACT

In this paper, we study a constrained utility maximization problem following the convex duality approach. After formulating the primal and dual problems, we construct the necessary and sufficient conditions for both the primal and dual problems in terms of forward and backward stochastic differential equations (FBSDEs) plus some additional conditions. Such formulation then allows us to explicitly characterize the primal optimal control as a function of the adjoint process coming from the dual FBSDEs in a dynamic fashion and vice versa. We also find that the optimal wealth process coincides with the adjoint process of the dual problem and vice versa. Finally we solve three constrained utility maximization problems, which contrasts the simplicity of the duality approach we propose and the technical complexity of solving the primal problem directly.     

Towards Strong Duality in Integer Programming

We consider in this paper the Lagrangian dual method for solving general integer programming. New properties of Lagrangian duality are derived by a means of perturbation analysis. In particular, a necessary and sufficient condition for a primal optimal solution to be generated by the Lagrangian relaxation is obtained. The solution properties of Lagrangian relaxation problem are studied systematically. To overcome the difficulties caused by duality gap between the primal problem and the dual problem, we introduce an equivalent reformulation for the primal problem via applying a pth power to the constraints. We prove that this reformulation possesses an asymptotic strong duality property. Primal feasibility and primal optimality of the Lagrangian relaxation problems can be achieved in this reformulation when the parameter p is larger than a threshold value, thus ensuring the existence of an optimal primal-dual pair. We further show that duality gap for this partial pth power reformulation is a strictly decreasing function of p in the case of a single constraint. Dedicated to Professor Alex Rubinov on the occasion of his 65th birthday. Research supported by the Research Grants Council of Hong Kong under Grant CUHK 4214/01E, and the National Natural Science Foundation of China under Grants 79970107 and 10571116.     

Duality for multiobjective fractional control problems with generalized invexity

Wolfe and Mond-Weir type duals for multiobjective control problems are formulated. Under pseudo-invexity/quasi-invexity assumptions on the functions involved, weak and strong duality theorems are proved to relate efficient solutions of the primal and dual problems.     

多目标分数变分问题的对偶性

本文利用参数逼近在函数广义（F，ρ）－凸的条件下，建立了一类多目标式变分问题关于有效解的对偶理论。