Optimization over the efficient set: overview

Over the past several decades, the optimization over the efficient set has seen a substantial development. The aim of this paper is to provide a state-of-the-art survey of the development. Given p linear criteria c ¹x,,c^p x and a feasible region X of R ⁿ, the linear multicriteria problem is to find a point x of X such that no point x' of X satisfies (c¹ x',,c^p x')(c¹ x,,c^p x) and (c¹x',,c^p x')q (c¹ x ,,c^p x). Such a point is called an efficient point. The optimization over the efficient set is the maximization of a given function over the set of efficient points. The difficulty of this problem is mainly due to the nonconvexity of this set. The existing algorithms for solving this problem could be classified into several groups such as adjacent vertex search algorithm, nonadjacent vertex search algorithm, branch-and-bound based algorithm, Lagrangian relaxation based algorithm, dual approach and bisection algorithm. In this paper we review a typical algorithm from each group and compare them from the computational point of view.     

Global Optimization in Practice: An Application to Interactive Multiple Objective Linear Programming

A multiple objective linear programming problem (P) involves the simultaneous maximization of two or more conflicting linear objective functions over a nonempty polyhedron X. Many of the most popular methods for solving this type of problem, including many well-known interactive methods, involve searching the efficient set X _E of the problem. Generally, however, X _E is a complicated, nonconvex set. As a result, concepts and methods from global optimization may be useful in searching X _E. In this paper, we will explain in theory, and show via an actual application to citrus rootstock selection in Florida, how the potential usefulness of the well-known interactive method STEM for solving problem (P) in this way, can depend crucially upon how accurately certain global optimization problems involving minimizations over X _E are solved. In particular, we will show both in theory and in practice that the choice of whether to use the popular but unreliable payoff table approach or to use one of the lesser known, more accurate global optimization methods to solve these problems can determine whether STEM succeeds or fails as a decision aid. Several lessons and conclusions of transferable value derived from this research are also given.     

Simplicially-Constrained DC Optimization over Efficient and Weakly Efficient Sets

We formulate optimization problems over efficient and weakly efficient sets as DC problems over a simplex in the criteria space. This formulation allows developing a decomposition algorithm using an adaptive simplex subdivision and a convex envelope function for solving both problems. Randomly generated problems up to the size of 150 decision variables and 7 criteria are solved.     

Minimum Maximal Flow Problem: An Optimization over the Efficient Set

The network flow theory and algorithms have been developed on the assumption that each arc flow is controllable and we freely raise and reduce it. We however consider in this paper the situation where we are not able or allowed to reduce the given arc flow. Then we may end up with a maximal flow depending on the initial flow as well as the way of augmentation. Therefore the minimum of the flow values that are attained by maximal flows will play an important role to see how inefficiently the network can be utilized. We formulate this problem as an optimization over the efficient set of a multicriteria program, propose an algorithm, prove its finite convergence, and report on some computational experiments.     

A Bilinear Algorithm for Optimizing a Linear Function over the Efficient Set of a Multiple Objective Linear Programming Problem

The problem Q of optimizing a linear function over the efficient set of a multiple objective linear program serves several useful purposes in multiple criteria decision making. However, Q is in itself a difficult global optimization problem, whose local optima, frequently large in number, need not be globally optimal. Indeed, this is due to the fact that the feasible region of Q is, in general, a nonconvex set. In this paper we present a monotonically increasing algorithm that finds an exact, globally-optimal solution for Q. Our approach does not require any hypothesis on the boundedness of neither the efficient set EP nor the optimal objective value. The proposed algorithm relies on a simplified disjoint bilinear program that can be solved through the use of well-known specifically designed methods within nonconvex optimization. The algorithm has been implemented in C and preliminary numerical results are reported.     

Branch-and-Bound Variant of an Outcome-Based Algorithm for Optimizing over the Efficient Set of a Bicriteria Linear Programming Problem

The paper presents a finite branch-and-bound variant of an outcome-based algorithm proposed by Benson and Lee for minimizing a lower-semicontinuous function over the efficient set of a bicriteria linear programming problem. Similarly to the Benson-Lee algorithm, we work primarily in the outcome space. Dissimilarly, instead of constructing a sequence of consecutive efficient edges in the outcome space, we use the idea of generating a refining sequence of partitions covering the at most two-dimensional efficient set in the outcome space. Computational experience is also presented.     

Conical Algorithm in Global Optimization for Optimizing over Efficient Sets

The problem of optimizing some contiuous function over the efficient set of a multiple objective programming problem can be formulated as a nonconvex global optimization problem with special structure. Based on the conical branch and bound algorithm in global optimization, we establish an algorithm for optimizing over efficient sets and discuss about the implementation of this algorithm for some interesting special cases including the case of biobjective programming problems.     

An Inner Approximation Method for Optimization over the Weakly Efficient Set

In this paper, we consider an optimization problem which aims to minimize a convex function over the weakly efficient set of a multiobjective programming problem. To solve such a problem, we propose an inner approximation algorithm, in which two kinds of convex subproblems are solved successively. These convex subproblems are fairly easy to solve and therefore the proposed algorithm is practically useful. The algorithm always terminates after finitely many iterations by compromising the weak efficiency to a multiobjective programming problem. Moreover, for a subproblem which is solved at each iteration of the algorithm, we suggest a procedure for eliminating redundant constraints.     

An Outer Approximation Algorithm for Generating All Efficient Extreme Points in the Outcome Set of a Multiple Objective Linear Programming Problem

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, some researchers in recent years have suggested that outcome set-based approaches should instead be developed and used to solve problem (MOLP). In this article, we present a finite algorithm, called the Outer Approximation Algorithm, for generating the set of all efficient extreme points in the outcome set of problem (MOLP). To our knowledge, the Outer Approximation Algorithm is the first algorithm capable of generating this set. As a by-product, the algorithm also generates the weakly efficient outcome set of problem (MOLP). Because it works in the outcome set rather than in the decision set of problem (MOLP), the Outer Approximation Algorithm has several advantages over decision set-based algorithms. It is also relatively easy to implement. Preliminary computational results for a set of randomly-generated problems are reported. These results tangibly demonstrate the usefulness of using the outcome set approach of the Outer Approximation Algorithm instead of a decision set-based approach. This revised version was published online in July 2006 with corrections to the Cover Date.     

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.     

On Efficient Solutions in Vector Optimization

A new characterization is obtained for the existence of an efficient solution of a vector optimization problem in terms of associated scalar optimization problems. The consequences for linear vector optimization problems are derived as a special case, Applications to convex vector optimization problems are also discussed.     

Primal and dual algorithms for optimization over the efficient set

ABSTRACT

Optimization over the efficient set of a multi-objective optimization problem is a mathematical model for the problem of selecting a most preferred solution that arises in multiple criteria decision-making to account for trade-offs between objectives within the set of efficient solutions. In this paper, we consider a particular case of this problem, namely that of optimizing a linear function over the image of the efficient set in objective space of a convex multi-objective optimization problem. We present both primal and dual algorithms for this task. The algorithms are based on recent algorithms for solving convex multi-objective optimization problems in objective space with suitable modifications to exploit specific properties of the problem of optimization over the efficient set. We first present the algorithms for the case that the underlying problem is a multi-objective linear programme. We then extend them to be able to solve problems with an underlying convex multi-objective optimization problem. We compare the new algorithms with several state of the art algorithms from the literature on a set of randomly generated instances to demonstrate that they are considerably faster than the competitors.     

Semivectorial Bilevel Optimization Problem: Penalty Approach

We consider a bilevel optimization problem where the upper level is a scalar optimization problem and the lower level is a vector optimization problem. For the lower level, we deal with weakly efficient solutions. We approach our problem using a suitable penalty function which vanishes over the weakly efficient solutions of the lower-level vector optimization problem and which is nonnegative over its feasible set. Then, we use an exterior penalty method.Communicated by H. P. Benson(Formerly Serban Bolintinéanu) Professor, University of New Caledonia, ERIM, Nouméa, New Caledonia. This author thanks the University of Naples Federico II for its support and the Department of Mathematics and Statistics for its hospitality.     

赋范线性空间中锥拟凸向量优化问题超有效解集的连通性

本文研究赋范线性空间中集值映射向量优化问题超有效解集的连通性问题．证明了目标映射为锥拟凸的向量优化问题的超有效解集是连通的．     

一类带反凸约束的非线性比式和问题的全局优化算法

本文针对一类带有反凸约束的非线性比式和分式规划问题,提出一种求其全局最优解的单纯形分支和对偶定界算法.该算法利用Lagrange对偶理论将其中关键的定界问题转化为一系列易于求解的线性规划问题.收敛性分析和数值算例均表明提出的算法是可行的.     

On Connectedness of Efficient Solution Sets for a Star Cone-Quasiconvex Vector Optimization Problem

In this note,we prove that the efficient solution set for a vector optimization problem with acontinuous,star cone-quasiconvex objective mapping is connected under the assumption that the ordering coneis a D-cone.A D-cone includes any closed convex pointed cones in a normed space which admits strictly positivecontinuous linear functionals.     

锥—连续凸映射的严有效解连通性

本文介绍了严有效解的概念,并在[1]工作的基础上进一步研究了锥连续拟凸映射的严有效的连通性。     

线性分式规划最优解集的求法

本文使用多面集的表示定理，导出了线性分式规划最优解集的结构，并给出确定全部最优解的计算步骤。     

On the connectedness of the set of weakly efficient points of a vector optimization problem in locally convex spaces

In vector optimization, topological properties of the set of efficient and weakly efficient points are of interest. In this paper, we study the connectedness of the setE ^w of all weakly efficient points of a subsetZ of a locally convex spaceX with respect to a continuous mappingp:X Y,Y locally convex and partially ordered by a closed, convex cone with nonempty interior. Under the general assumptions thatZ is convex and closed and thatp is a pointwise quasiconvex mapping (i.e., a generalized quasiconvex concept), the setE ^w is connected, if the lower level sets ofp are compact. Furthermore, we show some connectedness results on the efficient points and the efficient and weakly efficient outcomes. The considerations of this paper extend the previous results of Refs. 1–3. Moreover, some examples in vector approximation are given.The author is grateful to Dr. D. T. Luc and to a referee for pointing out an error in an earlier version of this paper.