Strong Duality for Generalized Convex Optimization Problems

In this paper, strong duality for nearly-convex optimization problems is established. Three kinds of conjugate dual problems are associated to the primal optimization problem: the Lagrange dual, Fenchel dual, and Fenchel-Lagrange dual problems. The main result shows that, under suitable conditions, the optimal objective values of these four problems coincide. The first author was supported in part by Gottlieb Daimler and Karl Benz Stiftung 02-48/99. This research has been performed while the second author visited Chemnitz University of Technology under DAAD (Deutscher Akademischer Austauschdienst) Grant A/02/12866. Communicated by T. Rapcsák     

Duality for almost convex optimization problems via the perturbation approach

We deal with duality for almost convex finite dimensional optimization problems by means of the classical perturbation approach. To this aim some standard results from the convex analysis are extended to the case of almost convex sets and functions. The duality for some classes of primal-dual problems is derived as a special case of the general approach. The sufficient regularity conditions we need for guaranteeing strong duality are proved to be similar to the ones in the convex case. The research of the first and third authors was partially supported by DFG (German Research Foundation), project WA 922/1. The research of the second author was supported by the grant PN II, ID 523/2007.     

Duality for location problems with unbounded unit balls

Given an optimization problem with a composite of a convex and componentwise increasing function with a convex vector function as objective function, by means of the conjugacy approach based on the perturbation theory, we determine a dual to it. Necessary and sufficient optimality conditions are derived using strong duality. Furthermore, as special case of this problem, we consider a location problem, where the “distances” are measured by gauges of closed convex sets. We prove that the geometric characterization of the set of optimal solutions for this location problem given by Hinojosa and Puerto in a recently published paper can be obtained via the presented dual problem. Finally, the Weber and the minmax location problems with gauges are given as applications.     

一类非光滑优化问题的最优性与对偶

本文研究了一类带等式和不等式约束的非光滑多目标优化问题,给出了该类问题的Karush-Kuhn-Tucker最优性必要条件和充分条件,建立了该类规划问题的一类混合对偶模型的弱对偶定理、强对偶定理、逆对偶定理、严格逆对偶定理和限制逆对偶定理.     

一类非光滑优化问题的最优性与对偶（英文）

本文研究了一类带等式和不等式约束的非光滑多目标优化问题,给出了该类问题的Karush-Kuhn-Tucker最优性必要条件和充分条件,建立了该类规划问题的一类混合对偶模型的弱对偶定理、强对偶定理、逆对偶定理、严格逆对偶定理和限制逆对偶定理.     

Duality on a nondifferentiable minimax fractional programming

We establish the necessary and sufficient optimality conditions on a nondifferentiable minimax fractional programming problem. Subsequently, applying the optimality conditions, we constitute two dual models: Mond-Weir type and Wolfe type. On these duality types, we prove three duality theorems??weak duality theorem, strong duality theorem, and strict converse duality theorem.     

Duality for nondifferentiable minimax programming in complex spaces

Employing the optimality (necessary and sufficient) conditions of a nondifferentiable minimax programming problem in complex spaces, we formulate a one-parametric dual and a parameter free dual problems. On both dual problems, we establish three duality theorems: weak, strong, and strict converse duality theorem, and prove that there is no duality gap between the two dual problems with respect to the primal problem under some generalized convexities of complex functions in the complex programming problem.     

Duality for nondifferentiable minimax programming in complex spaces

Employing the optimality (necessary and sufficient) conditions of a nondifferentiable minimax programming problem in complex spaces, we formulate a one-parametric dual and a parameter free dual problems. On both dual problems, we establish three duality theorems: weak, strong, and strict converse duality theorem, and prove that there is no duality gap between the two dual problems with respect to the primal problem under some generalized convexities of complex functions in the complex programming problem.     

Multiobjective Control Approximation Problems: Duality and Optimality

A general convex multiobjective control approximation problem is considered with respect to duality. The single objectives contain linear functionals and powers of norms as parts, measuring the distance between linear mappings of the control variable and the state variables. Moreover, linear inequality constraints are included. A dual problem is established, and weak and strong duality properties as well as necessary and sufficient optimality conditions are derived. Point-objective location problems and linear vector optimization problems turn out to be special cases of the problem investigated. Therefore, well-known duality results for linear vector optimization are obtained as special cases.     

Optimality Criteria and Duality in Multiobjective Programming Involving Nonsmooth Invex Functions

In this paper a generalization of invexity is considered in a general form, by means of the concept of K-directional derivative. Then in the case of nonlinear multiobjective programming problems where the functions involved are nondifferentiable, we established sufficient optimality conditions without any convexity assumption of the K-directional derivative. Then we obtained some duality results.     

Further Study on Augmented Lagrangian Duality Theory

In this paper, we present a necessary and sufficient condition for a zero duality gap between a primal optimization problem and its generalized augmented Lagrangian dual problems. The condition is mainly expressed in the form of the lower semicontinuity of a perturbation function at the origin. For a constrained optimization problem, a general equivalence is established for zero duality gap properties defined by a general nonlinear Lagrangian dual problem and a generalized augmented Lagrangian dual problem, respectively. For a constrained optimization problem with both equality and inequality constraints, we prove that first-order and second-order necessary optimality conditions of the augmented Lagrangian problems with a convex quadratic augmenting function converge to that of the original constrained program. For a mathematical program with only equality constraints, we show that the second-order necessary conditions of general augmented Lagrangian problems with a convex augmenting function converge to that of the original constrained program.This research is supported by the Research Grants Council of Hong Kong (PolyU B-Q359.)     

Conjugate Duality and the Control of Linear Discrete Systems

In this paper we deal with the minimization of a convex function over the solution set of a range inclusion problem determined by a multivalued operator with convex graph. We attach a dual problem to it, provide regularity conditions guaranteeing strong duality and derive for the resulting primal–dual pair necessary and sufficient optimality conditions. We also discuss the existence of optimal solutions for the primal and dual problems by using duality arguments. The theoretical results are applied in the context of the control of linear discrete systems.     

Duality theorem of nondifferentiable convex multiobjective programming

Necessary and sufficient conditions of Fritz John type for Pareto optimality of multiobjective programming problems are derived. This article suggests to establish a Wolfe-type duality theorem for nonlinear, nondifferentiable, convex multiobjective minimization problems. The vector Lagrangian and the generalized saddle point for Pareto optimality are studied. Some previously known results are shown to be special cases of the results described in this paper.This research was partly supported by the National Science Council, Taipei, ROC.The authors would like to thank the two referees for their valuable suggestions on the original draft.     

Optimality Conditions and Duality for a Class of Nonlinear Fractional Programming Problems

In this paper, we present sufficient optimality conditions and duality results for a class of nonlinear fractional programming problems. Our results are based on the properties of sublinear functionals and generalized convex functions.     

Fractional multi-commodity flow problem: Duality and optimality conditions

This paper deals with multi-commodity flow problem with fractional objective function. The optimality conditions and the duality concepts of this problem are given. For this aim, the fractional linear programming formulation of this problem is considered and the weak duality, the strong direct duality and the weak complementary slackness theorems are proved applying the traditional duality theory of linear programming problems which is different from same results in Chadha and Chadha (2007) [1]. In addition, a strong (strict) complementary slackness theorem is derived which is firstly presented based on the best of our knowledge. These theorems are transformed in order to find the new reduced costs for fractional multi-commodity flow problem. These parameters can be used to construct some algorithms for considered multi-commodity flow problem in a direct manner. Throughout the paper, the boundedness of the primal feasible set is reduced to a weaker assumption about solvability of primal problem which is another contribution of this paper. Finally, a real world application of the fractional multi-commodity flow problem is presented.     

Optimality and Duality for a Class of Nondifferentiable Multiobjective Fractional Programming Problems

In this paper, we consider a class of nondifferentiable multiobjective fractional programs in which each component of the objective function contains a term involving the support function of a compact convex set. We establish necessary and sufficient optimality conditions and duality results for weakly efficient solutions of nondifferentiable multiobjective fractional programming problems. This work was supported by Grant R01-2003-000-10825-0 from the Basic Research Program of KOSEF.     

非凸分式优化问题的最优性充分条件和对偶

通过引入广义弧连通概念,在Rn空间中,研究极大极小非凸分式规划问题的最优性充分条件及其对偶问题.首先获得了极大极小非凸分式规划问题的最优性充分条件;然后建立分式规划问题的一个对偶模型并得到了弱对偶定理,强对偶定理和逆对偶定理.     

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.     

Parameter-Free Dual Models for Fractional Programming with Generalized Invexity

By parameter-free approach, we establish sufficient optimality conditions for nondifferentiable fractional variational programming under certain specific structure of generalized invexity. Employing the sufficient optimality conditions, two parameter-free dual models are formulated. The weak duality, strong duality and strict converse duality theorems are proved in the framework of generalized invexity.     

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.