群体多目标规划的群体一致有效公理

本文研究使用单个多目标规划来替代群体多目标规划（Ｇｒｏｕｐ Ｍｕｌｔｉｏｂｊｅｃｔｉｖｅ Ｐｒｏｇｒａｍ ｍ ｉｎｇ）（ＧＭＰ）的公理问题．     

多目标规划弱较多有效解的充要条件

本文构造欧氏空间上的一个连续泛函，建立关于非凸锥H的分离定理，给出多目标规划弱较多有效解的充要条件．     

多目标分式规划的两种新对偶形式

§1.引言和引理 在[1]和[2]中,C.Singh和林锉云曾分别研究了多目标分式规划的对偶问题,本文则给出多目标分式规划的另外两种新的对偶形式。这两种对偶规划和R.Jagannathan以及C.Bector关于非线性规划的对偶理论有关。最后,我们还讨论了所研究的两种多目标分式对偶规划之间的相互关系。 考虑多目标分式规划     

集值映射多目标规划问题的解集的连通性

本文研究了无限维空间里集值映射的多目标规划问题。主要结果是：当目标函数为类凸的集值映射时，其目标空间里的有效点集是连通的；若目标函数为Ｃ－凸的集值映射时，其有效解集也连通。     

多目标规划ak-较多有效解类的有效性充分条件

在文［３］引进多目标规划问题的带双参数ａ和ｋ的ａｋ－较多有效解, 并且给出了此类解的有效性必要条件。本文在一定凸性的假设下证明了ａｋ－较多 有效解和ａｋ－弱较多有效解的几个有效性充分条件．     

多目标规划的对偶二阶条件

本文针对多目标规划 ( VP)的 Lagrange对偶规划 ( VD) ,从几何直观的角度出发 ,给出对偶规划( VD)的二阶最优性条件 ,即对偶二阶条件 ,并证明了相应的最优性定理 .     

二层多目标规划的最优性条件

本文在集值优化的框架下提出了一个二层多目标规划模型（BLMOP）.利用集值映射的相依导数和相依上导数,给出了几个有关（BLMOP）的弱有效解的必要或充分最优性条件.     

多目标规划的圆锥有效解

本文利用有限维向量空间中圆锥的概念,引入了多目标规划问题的一种新的有效解－圆锥有效解,并讨论了这种有效解的性质。同时,讲座了圆锥有效解与Pareto有效解以及绝对最优解之间的关系。最后,通过引进目标总值差异概念,分析了圆锥有效解的主要特点。     

锥连续锥拟凹多目标规划问题弱有效解集的连通性

本文讨论了锥连续锥拟凹向量函数的弱有效解集的连通性问题；证明了当可行集X为紧凸集，目标函数f在X上锥连续锥拟凹时，其弱有效解集是连通的，在一定的意义下，改进了文献[1] 中的结果。     

Measuring the quality of discrete representations of efficient sets in multiple objective mathematical programming

One way of solving multiple objective mathematical programming problems is finding discrete representations of the efficient set. A modified goal of finding good discrete representations of the efficient set would contribute to the practicality of vector maximization algorithms. We define coverage, uniformity and cardinality as the three attributes of quality of discrete representations and introduce a framework that includes these attributes in which discrete representations can be evaluated, compared to each other, and judged satisfactory or unsatisfactory by a Decision Maker. We provide simple mathematical programming formulations that can be used to compute the coverage error of a given discrete representation. Our formulations are practically implementable when the problem under study is a multiobjective linear programming problem. We believe that the interactive algorithms along with the vector maximization methods can make use of our framework and its tools. Received April 7, 1998 / Revised version received March 1999?Published online November 9, 1999     

Linear complementarity problems and multiple objective programming

An equivalence is demonstrated between solving a linear complementarity problem with general data and finding a certain subset of the efficient points of a multiple objective programming problem. A new multiple objective programming based approach to solving linear complementarity problems is presented. Results on existence, uniqueness and computational complexity are included.     

A geometrical analysis of the efficient outcome set in multiple objective convex programs with linear criterion functions

This article performs a geometrical analysis of the efficient outcome setY _E of a multiple objective convex program (MLC) with linear criterion functions. The analysis elucidates the facial structure ofY _E and of its pre-image, the efficient decision setX _E . The results show thatY _E often has a significantly-simpler structure thanX _E . For instance, although both sets are generally nonconvex and their maximal efficient faces are always in one-to-one correspondence, large numbers of extreme points and faces inX _E can map into non-facial subsets of faces inY _E , but not vice versa. Simple tests for the efficiency of faces in the decision and outcome sets are derived, and certain types of faces in the decision set are studied that are immune to a common phenomenon called collapsing. The results seem to indicate that significant computational benefits may potentially be derived if algorithms for problem (MLC) were to work directly with the outcome set of the problem to find points and faces ofY _E , rather than with the decision set.     

A compromise procedure for the multiple objective linear fractional programming problem

A generalization of a well-known multiple objective linear fractional programming (MOLFP) problem, the multiple objective fractional programming (MOFP) problem, is formulated. A concept of multiple objective programming (MOP) problem corresponding to MOFP is introduced and some relations between those problems are examined. Based on these results, a compromise procedure for MOLFP problem is proposed. A numerical example is given to show how the procedure works.     

An all-linear programming relaxation algorithm for optimizing over the efficient set

The problem (P) of optimizing a linear function over the efficient set of a multiple objective linear program has many important applications in multiple criteria decision making. Since the efficient set is in general a nonconvex set, problem (P) can be classified as a global optimization problem. Perhaps due to its inherent difficulty, it appears that no precisely-delineated implementable algorithm exists for solving problem (P) globally. In this paper a relaxation algorithm is presented for finding a globally optimal solution for problem (P). The algorithm finds an exact optimal solution to the problem after a finite number of iterations. A detailed discussion is included of how to implement the algorithm using only linear programming methods. Convergence of the algorithm is proven, and a sample problem is solved.Research supported by a grant from the College of Business Administration, University of Florida, Gainesville, Florida, U.S.A.     

Random problem generation and the computation of efficient extreme points in multiple objective linear programming

This paper looks at the task of computing efficient extreme points in multiple objective linear programming. Vector maximization software is reviewed and the ADBASE solver for computing all efficient extreme points of a multiple objective linear program is described. To create MOLP test problems, models for random problem generation are discussed. In the computational part of the paper, the numbers of efficient extreme points possessed by MOLPs (including multiple objective transportation problems) of different sizes are reported. In addition, the way the utility values of the efficient extreme points might be distributed over the efficient set for different types of utility functions is investigated. Not surprisingly, results show that it should be easier to find good near-optimal solutions with linear utility functions than with, for instance, Tchebycheff types of utility functions.Dedicated to Professor George B. Dantzig on the occasion of his eightieth birthday.