Outcome Space Partition of the Weight Set in Multiobjective Linear Programming

Approaches for generating the set of efficient extreme points of the decision set of a multiple-objective linear program (P) that are based upon decompositions of the weight set W⁰ suffer from one of two special drawbacks. Either the required computations are redundant, or not all of the efficient extreme point set is found. This article shows that the weight set for problem (P) can be decomposed into a partition based upon the outcome set Y of the problem, where the elements of the partition are in one-to-one correspondence with the efficient extreme points of Y. As a result, the drawbacks of the decompositions of W⁰ based upon the decision set of problem (P) disappear. The article explains also how this new partition offers the potential to construct algorithms for solving large-scale applications of problem (P) in the outcome space, rather than in the decision space.     

Multiple-objective approximation of feasible but unsolvable linear complementarity problems

Using a multiple-objective framework, feasible linear complementarity problems (LCPs) are handled in a unified manner. The resulting procedure either solves the feasible LCP or, under certain conditions, produces an approximate solution which is an efficient point of the associated multiple-objective problem. A mathematical existence theory is developed which allows both specialization and extension of earlier results in multiple-obkective programming. Two perturbation approaches to finding the closest solvable LCPs to a given unsolvable LCP are proposed. Several illustrative examples are provided and discussed.     

Further Analysis of an Outcome Set-Based Algorithm for Multiple-Objective Linear Programming

Various difficulties have been encountered in using decision set-based vector maximization methods to solve a multiple-objective linear programming problem (MOLP). Motivated by these difficulties, Benson recently developed a finite, outer-approximation algorithm for generating the set of all efficient extreme points in the outcome set, rather than in the decision set, of problem (MOLP). In this article, we show that the Benson algorithm also generates the set of all weakly efficient points in the outcome set of problem (MOLP). As a result, the usefulness of the algorithm as a decision aid in multiple objective linear programming is further enhanced.     

Solving sum-of-ratios fractional programs using efficient points

Constrained maximization of a sum of p1 ratios is a difficult nonconvex optimization problem (even if all functions involved are linear) with many applications in management sciences. In this paper, we first give a brief introductory survey of this problem. Then we propose a general branch-and-bound algorithm which uses rectangular partitions in the Euclidean space of dimension p. Theoretically, this algorithm is applicable under very general assumptions. Practically, we give an efficient implementation for fine fractions. Here the bounding procedures use dual constructions and the calculation of efficient points of a corresponding multiple-objective optimization problem. Finally, we present some promising numerical results     

Hybrid Approach for Solving Multiple-Objective Linear Programs in Outcome Space

Various difficulties arise in using decision set-based vector maximization methods to solve a multiple-objective linear programming problem (MOLP). Motivated by these difficulties, some researchers in recent years have begun to develop tools for analyzing and solving problem (MOLP) in outcome space, rather than in decision space. In this article, we present and validate a new hybrid vector maximization approach for solving problem (MOLP) in outcome space. The approach systematically integrates a simplicial partitioning technique into an outer approximation procedure to yield an algorithm that generates the set of all efficient extreme points in the outcome set of problem (MOLP) in a finite number of iterations. Some key potential practical and computational advantages of the approach are indicated.     

Utility Function Programs and Optimization over the Efficient Set in Multiple-Objective Decision Making

Natural basic concepts in multiple-objective optimization lead to difficult multiextremal global optimization problems. Examples include detection of efficient points when nonconvexities occur, and optimization of a linear function over the efficient set in the convex (even linear) case. Assuming that a utility function exists allows one to replace in general the multiple-objective program by a single, nonconvex optimization problem, which amounts to a minimization over the efficient set when the utility function is increasing. A new algorithm is discussed for this utility function program which, under natural mild conditions, converges to an -approximate global solution in a finite number of iterations. Applications include linear, convex, indefinite quadratic, Lipschitz, and d.c. objectives and constraints.     

Solution of linear complementarity problems using minimization with simple bounds

We define a minimization problem with simple bounds associated to the horizontal linear complementarity problem (HLCP). When the HLCP is solvable, its solutions are the global minimizers of the associated problem. When the HLCP is feasible, we are able to prove a number of properties of the stationary points of the associated problem. In many cases, the stationary points are solutions of the HLCP. The theoretical results allow us to conjecture that local methods for box constrained optimization applied to the associated problem are efficient tools for solving linear complementarity problems. Numerical experiments seem to confirm this conjecture.This work was supported by FAPESP (grants 90-3724-6 and 91-2441-3), CNPq and FAEP (UNICAMP).     

A revised simplex method for linear multiple objective programs

For linear multiple-objective problems, a necessary and sufficient condition for a point to be efficient is employed in the development of a revised simplex algorithm for the enumeration of the set of efficient extreme points. Five options within this algorithm were tested on a variety of problems. Results of these tests provide indications for effective use of the algorithm.This work was partially supported by the Office of Naval Research, Contract No. N00014-67-A-0321-0003 (NR047-095).     

Existence of Efficient Points in Vector Optimization and Generalized Bishop–Phelps Theorem

In a set without linear structure equipped with a preorder, we give a general existence result for efficient points. In a topological vector space equipped with a partial order induced by a closed convex cone with a bounded base, we prove another kind of existence result for efficient points; this result does not depend on the Zorn lemma. As applications, we study a solution problem in vector optimization and generalize the Bishop–Phelps theorem to a topological vector space setting by showing that the B-support points of any sequentially complete closed subset A of a topological vector space E is dense in A, where B is any bounded convex subset of E.     

A finite,nonadjacent extreme-point search algorithm for optimization over the efficient set

The problem (P) of optimizing a linear function over the efficient set of a multiple-objective linear program serves many useful purposes in multiple-criteria decision making. Mathematically, problem (P) can be classified as a global optimization problem. Such problems are much more difficult to solve than convex programming problems. In this paper, a nonadjacent extreme-point search algorithm is presented for finding a globally optimal solution for problem (P). The algorithm finds an exact extreme-point optimal solution for the problem after a finite number of iterations. It can be implemented using only linear programming methods. Convergence of the algorithm is proven, and a discussion is included of its main advantages and disadvantages.The author owes thanks to two anonymous referees for their helpful comments.     

Inverse DEA under inter-temporal dependence using multiple-objective programming

This paper deals with the inverse Data Envelopment Analysis (DEA) under inter-temporal dependence assumption. Both problems, input-estimation and output-estimation, are investigated. Necessary and sufficient conditions for input/output estimation are established utilizing Pareto and weak Pareto solutions of linear multiple-objective programming problems. Furthermore, in this paper we introduce a new optimality notion for multiple-objective programming problems, periodic weak Pareto optimality. These solutions are used in inverse DEA, and it is shown that these can be characterized by a simple modification in weighted sum scalarization tool.     

A methodology for assessing eco-efficiency in logistics networks

Recent literature on sustainable logistics networks points to two important questions: (i) How to spot the preferred solution(s) balancing environmental and business concerns? (ii) How to improve the understanding of the trade-offs between these two dimensions? We posit that a visual exploration of the efficient frontier and trade-offs between profitability and environmental impacts are particularly suitable to answer these two questions. The visual representation of the efficient frontier, however, presents two challenges. The first is to obtain a good approximation for such frontier without enumerating all extreme efficient solutions. The second is to obtain a good visual representation of the efficient frontier. We propose a two-phased heuristic to handle these two problems. The algorithm is designed for the multi-objective linear problem with three objectives: minimize costs, cumulative energy demand and waste in a reverse logistics network. We illustrate our approach by designing a complex recycling logistics network in Germany.     

赋范线性空间集合的严有效点

本文引入一个新的有效点概念一严有效点,它是Borwein超有效点的推广.此外还讨论了严有效点的基本性质:存在性条件、纯量化特征、稠密性定理以及与Borwein超有效点的关系.     

Multiplicative Programming Problems: Analysis and Efficient Point Search Heuristic

Multiplicative programming problems are difficult global optimization problems known to be NP-hard. At the same time, these problems have some important applications in engineering, finance, economics, and other fields. This article has two purposes. The first is to present an analysis that shows several relationships between concave multiplicative programs and concave minimization problems, and between concave multiplicative programs and certain multiple-objective mathematical programs. The second purpose is to propose and report computational results for a heuristic efficient-point search algorithm that we have designed for use on linear multiplicative programming problems. To our knowledge, this is the first heuristic algorithm of its type. The theoretical and algorithmic results given in the article offer some potentially important new avenues for analyzing and solving multiplicative programming problems of various types.     

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.     

An Algorithm for Solving the Minimum-Norm Point Problem over the Intersection of a Polytope and an Affine Set

In this paper, we propose an efficient algorithm for finding the minimum-norm point in the intersection of a polytope and an affine set in an n-dimensional Euclidean space, where the polytope is expressed as the convex hull of finitely many points and the affine set is expressed as the intersection of k hyperplanes, k1. Our algorithm solves the problem by using directly the original points and the hyperplanes, rather than treating the problem as a special case of the general quadratic programming problem. One of the advantages of our approach is that our algorithm works as well for a class of problems with a large number (possibly exponential or factorial in n) of given points if every linear optimization problem over the convex hull of the given points is solved efficiently. The problem considered here is highly degenerate, and we take care of the degeneracy by solving a subproblem that is a conical version of the minimum-norm point problem, where points are replaced by rays. When the number k of hyperplanes expressing the affine set is equal to one, we can easily avoid degeneracy, but this is not the case for k2. We give a subprocedure for treating the degenerate case. The subprocedure is interesting in its own right. We also show the practical efficiency of our algorithm by computational experiments.     

Kuratowski convergence of the efficient sets in the parametric linear vector semi-infinite optimization

In the space of whole linear vector semi-infinite optimization problems we consider the mappings putting into correspondence to each problem the set of efficient and weakly efficient points, respectively. We endow the image space with Kuratowski convergence and by means of the lower and upper semi-continuity of these mappings we prove generic well-posedness of the vector optimization problems. The connection between the continuity and some properties of the efficient sets is also discussed.     

Comparison of Existence Results for Efficient Points

Existence results of maximal points with respect to general binary relations were stated by Hazen and Morin (Ref. 1) and by Gajek and Zagrodny (Ref. 2). In this paper, we point out that the natural framework for this problem is that of transitive and reflexive relations (preorders). The aim of this paper is to discuss existence results for maximal points with respect to general transitive relations in such a way that, when considering them for preorders defined by convex cones, we are able to recover most known existence results for efficient points; the quasi-totality of them, with their (short) proofs, is presented, too.     

Determination of the efficient set in multiobjective linear programming

This paper develops a method for finding the whole set of efficient points of a multiobjective linear problem. Two algorithms are presented; the first one describes the set of all efficient vertices and all efficient rays of the constraint polyhedron, while the second one generates the set of all efficient faces. The method has been tested on several examples for which numerical results are reported.The authors are grateful to Professor W. Stadler and an anonymous referee for their helpful comments and corrections.     

Efficient Solution Concepts and Their Relations in Stochastic Multiobjective Programming

In this work, different concepts of efficient solutions to problems of stochastic multiple-objective programming are analyzed. We center our interest on problems in which some of the objective functions depend on random parameters. The existence of different concepts of efficiency for one single stochastic problem, such as expected-value efficiency, minimum-risk efficiency, etc., raises the question of their quality. Starting from this idea, we establish some relationships between the different concepts. Our study enables us to determine what type of efficient solutions are obtained by each of these concepts.