Algorithms for solving multiobjective discrete control problems and dynamic c-games on networks

In this paper we study a special class of multiobjective discrete control problems on dynamic networks. We assume that the dynamics of the system is controlled by p actors (players) and each of them intend to minimize his own integral-time cost by a certain trajectory. Applying Nash and Pareto optimality principles we study multiobjective control problems on dynamic networks where the dynamics is described by a directed graph.Polynomial-time algorithms for determining the optimal strategies of the players in the considered multiobjective control problems are proposed exploiting the special structure of the underlying graph. Properties of time-expanded networks are characterized. A constructive scheme which consists of several algorithms is presented.     

Characterization of solutions of multiobjective optimization problem

A characterization of weakly efficient, efficient and properly efficient solutions of multiobjective optimization problems is given in terms of a scalar optimization problem by using a special “distance” function. The concept of the well-posedness for this special scalar problem is then linked with the properly efficient solutions of the multiobjective problem.     

A multiobjective fuzzy bandwidth partitioning model for self-sizing networks

The ability to cope with dynamic bandwidth demands is a special feature for Quality of Service provisioning in networks carrying bandwidth hungry applications. This paper introduces a novel approach based on multiobjective fuzzy optimization for dynamic bandwidth allocation. This new approach deals with uncertain bandwidth demands more efficiently than approaches based on Classical Optimization Theory and yet supports Quality of Service commitments.     

Hierarchical generating method for large-scale multiobjective systems

This paper investigates large-scale multiobjective systems in the context of a general hierarchical generating method which considers the problem of how to find the set of all noninferior solutions by decomposition and coordination. A new, unified framework of the hierarchical generating method is developed by integrating the envelope analysis approach and the duality theory that is used in multiobjective programming. In this scheme, the vector-valued Lagrangian and the duality theorem provide the basis of a decomposition of the overall multiobjective system into several multiobjective subsystems, and the envelope analysis gives an efficient approach to deal with the coordination at a high level. The following decomposition-coordination schemes for different problems are developed: (i) a spatial decomposition and envelope coordination algorithm for large-scale multiobjective static systems; (ii) a temporal decomposition and envelope coordination algorithm for multiobjective dynamic systems; and (iii) a three-level structure algorithm for large-scale multiobjective dynamic systems.This work was supported by NSF Grant No. CEE-82-11606.     

Fuzzy goal programming for multiobjective transportation problems

Several fuzzy approaches can be considered for solving multiobjective transportation problem. This paper presents a fuzzy goal programming approach to determine an optimal compromise solution for the multiobjective transportation problem. We assume that each objective function has a fuzzy goal. Also we assign a special type of nonlinear (hyperbolic) membership function to each objective function to describe each fuzzy goal. The approach focuses on minimizing the negative deviation variables from 1 to obtain a compromise solution of the multiobjective transportation problem. We show that the proposed method and the fuzzy programming method are equivalent. In addition, the proposed approach can be applied to solve other multiobjective mathematical programming problems. A numerical example is given to illustrate the efficiency of the proposed approach.     

信息集广义多目标博弈弱Pareto-Nash平衡点的存在性和稳定性

首先把信息集的概念引入到多目标博弈, 建立了信息集广义多目标博弈模型, 并指出了信息集广义多目标博弈以广义多目标博弈、广义n人非合作博弈、一般n人非合作博弈为特例, 然后用Fan-Glicksberg不动点定理证明了信息集广义多目标博弈弱Pareto-Nash平衡点的存在性, 最后在本质解和强本质解的意义下, 分别研究了信息集广义多目标博弈弱Pareto-Nash平衡点的通有稳定性和强本质连通区的存在性.     

具B-I凸性条件下变分控制问题的混合对偶性

在广义B-凸性条件下,建立了多目标变分控制问题的混合对偶模型,使得Mond-Weir型和Wolfe型对偶成为其特殊情况,并建立了关于有效解的混合对偶理论.     

New approach for nonseparable dynamic programming problems

A general class of nonseparable dynamic problems is studied in a dynamic programming framework by introducingkth-order separability. The solution approach uses multiobjective dynamic programming as a separation strategy forkth-order separable dynamic problems. The theoretical grounding on which the optimal solution of the original nonseparable dynamic problem can be attained by a noninferior solution of the corresponding multiobjective dynamic programming problem is established. The relationship between the overall optimal Lagrangian multipliers and the stage-optimal Lagrangian multipliers and the relationship between the overall weighting vector and the stage weighting vector are explored, providing the basis for identifying the optimal solution of the original nonseparable problem from among the set of noninferior solutions generated by the envelope approach.This work was supported in part by NSF Grant No. CES-86-17984. The authors appreciate the comments from Dr. V. Chankong and the editorial work by Mrs. V. Benade and Dr. S. Hitchcock.     

ProperD-solutions of multiobjective programming problems with set functions

A special class of solutions for multiobjective programming problems with set functions is considered. A subset of nondominated solutions, called properD-solution set, with respect to a given domination structure is characterized under two situations, with and without inequality constraints.The authors greatly appreciate valuable comments received from the referees.     

The existence of solutions for the system of generalized vector quasi-equilibrium problems

In this paper, we study the system of generalized vector quasi-equilibrium problems which includes as special cases the system of generalized vector equilibrium problems and the system of vector quasi-equilibrium problems, and establish the existence result for its solutions by means of the Kakutani–Fan–Glicksberg fixed points theorem. As special cases, we also derive the existence results for weakly Pareto–Nash equilibrium points for multiobjective generalized game problems and multiobjective game problems.     

Solving a multiobjective traveling salesman problem by dynamic programming

In this paper Klötzler's method of multiobjective dynamic programming is applied to the solution of a two-dimensional traveling salesman problem. In this way Bellman's and Held/Karp's dynamic programming approach to one-dimensional traveling salesman problems is extended to the multiobjective case.     

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.     

Characterization of nondominated controls in terms of solutions of weighting problems

The problem of continuously dynamic multiobjective optimization, or multiobjective control, is discussed. The concepts of nondominated controls and viable controls are introduced. For a class of nonlinear dynamic systems, the convexity of their controlled Carathéodory trajectories is proved. Based on this convexity, sufficient conditions are given for the solution of a multiobjective control problem to be obtained in terms of solutions of weighting problems.This work was partly done at the Huazhong University of Science and Technology, Wuhan, China. The author is indebted to Professor Chen Ting for his advice. The author wishes to thank Professor G. Leitmann for his favorable comments. The author is also grateful to Ms. Mary S. Linn at the University of Kansas, who helped him improve the English presentation of this paper.     

多目标变分问题的混合对偶性

本文给出了一类多目标变分问题的混合对偶 ,使得 Wolfe型对偶和 Mond-Weir型对偶是其特殊情况 ,并在函数 (F ,ρ) -凸性的条件下建立了多目标变分问题关于有效解的混合对偶理论 .     

Derived Sets for Weak Multiobjective Optimization Problems with State and Control Variables

The concept of a K-gradient, introduced in Ref. 1 in order to generalize the concept of a derived convex cone defined by Hestenes, is extended to weak multiobjective optimization problems including not only a state variable, but also a control variable. The new concept is employed to state multiplier rules for the local solutions of such dynamic multiobjective optimization problems. An application of these multiplier rules to the local solutions of an abstract multiobjective optimal control problem yields general necessary optimality conditions that can be used to derive concrete maximum principles for multiobjective optimal control problems, e.g., problems described by integral equations with additional functional constraints.     

On multiobjective generalized symmetric dual programs withρ − (η, θ)-invexity

A pair of multiobjective generalized symmetric dual nonlinear programming problems and weak, strong and converse duality theorems for these problems are established under generalized ρ ? (η, θ)-invexity assumptions. Several known results are obtained as special cases.     

Higher-Order Duality in Nondifferentiable Multiobjective Programming

A new generalized class of higher-order (F, α, ρ, d)–type I functions is introduced, and a general Mond–Weir type higher-order dual is formulated for a nondifferentiable multiobjective programming problem. Based on the concepts introduced, various higher-order duality results are established. At the end, some special cases are also discussed.     

Bundle-based descent method for nonsmooth multiobjective DC optimization with inequality constraints

Multiobjective DC optimization problems arise naturally, for example, in data classification and cluster analysis playing a crucial role in data mining. In this paper, we propose a new multiobjective double bundle method designed for nonsmooth multiobjective optimization problems having objective and constraint functions which can be presented as a difference of two convex (DC) functions. The method is of the descent type and it generalizes the ideas of the double bundle method for multiobjective and constrained problems. We utilize the special cutting plane model angled for the DC improvement function such that the convex and the concave behaviour of the function is captured. The method is proved to be finitely convergent to a weakly Pareto stationary point under mild assumptions. Finally, we consider some numerical experiments and compare the solutions produced by our method with the method designed for general nonconvex multiobjective problems. This is done in order to validate the usage of the method aimed specially for DC objectives instead of a general nonconvex method.     

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.