On a domination property for vector maximization with respect to cones

We give a corrected version of Theorem 2.1 of Ref. 1.The author is indebted to D. T. Luc, Hungarian Academy of Sciences, Budapest, Hungary, for pointing out his error in the proof of the original version of Theorem 2.1 in Ref. 1.     

Existence of efficient solutions for vector maximization problems

The vector maximization problem arises when more than one objective function is to be maximized over a given feasibility region. The concept of efficiency has played a useful role in analyzing this problem. In order to exclude efficient solutions of a certain anomalous type, the concept of proper efficiency has also been utilized. In this paper, an examination of the existence of efficient and properly efficient solutions for the vector maximization problem is undertaken. Given a feasible solution for the vector maximization problem, a related single-objective mathematical programming problem is investigated. Any optimal solution to this program, if one exists, yields an efficient solution for the vector maximization problem. In many cases, the unboundedness of this problem shows that no properly efficient solutions exist. Conditions are pointed out under which the latter conclusion implies that the set of efficient solutions is null. As a byproduct of our results, conditions are derived which guarantee that the outcome of any improperly efficient point is the limit of the outcomes of some sequence of properly efficient points. Examples are provided to illustrate these results.The author would like to thank Professor T. L. Morin for his helpful comments. Thanks also go to an anonymous reviewer for his useful comments concerning an earlier version of this paper.The author would like to acknowledge a useful discussion with Professor G. Bitran which helped in motivating Example 4.1.     

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.     

Generalized penalization and maximization of vectorial nonsmooth functions

We use directional Lipschitz concepts and a minimal time function with respect to a set of directions in order to derive generalized penalization results for Pareto minimality in set-valued constrained optimization. Then, we obtain necessary optimality conditions for maximization in constrained vector optimization in terms of generalized differentiation objects. To the latter aim, we deduce first some enhanced calculus rules for coderivatives of the difference of two mappings. All the main results of this paper are tailored to model directional features of the optimization problem under study.     

Inverse programming and the linear vector maximization problem

This paper presents a new concept of efficient solution for the linear vector maximization problem. Briefly, these solutions are efficient with respect to the constraints, in addition to being efficient with respect to the multiple objectives. The duality theory of linear vector maximization is developed in terms of this solution concept and then is used to formulate the problem as a linear program.This research has been partially supported by grants from the Canada Council and the National Research Council of Canada.     

On the domination property in vector optimization

An example is given to show the inadequacy of the result of Ref. 1 concerning the domination property of a convex vector maximization problem with respect to cones. A necessary and sufficient condition for the domination property to hold is supplied.     

Pareto Analysis vis-à-vis Balance Space Approach in Multiobjective Global Optimization

There is much controversy about the balance space approach, introduced first in Ref. 1, pp. 138–140, with the consideration of the balance number and balance vectors, and then further developed in Ref. 2, with the consideration of balance points and balance sets. There were attempts to identify the balance space approach with some other methods of multiobjective optimization, notably the method proposed in Ref. 3 and most recently Pareto analysis, as presented in Ref. 4. In this paper, we compare Pareto analysis with the balance space approach on several examples to demonstrate the interrelation and the differences of the two methods. As a byproduct, it is shown that, in some cases, the entire Pareto sets, proper and adjoint, can be determined very simply, without any special investigation of the (nonscalarized, nonconvex) multiobjective global optimization problem. The method of parameter introduction is presented in application to determining the Pareto sets and balance set. The use of computer graphics software complemented with the Gauss–Jordan matrix reduction algorithm is proposed for a class of otherwise intractable problems with nonconvex constraint sets.     

Radial Solutions and Orthogonal Trajectories in Multiobjective Global Optimization

This paper presents a new, ray-oriented method for the global solution of nonscalarized vector optimization problems and a framework for the application of the Karush–Kuhn–Tucker theorem to such problems. Properties of nonlinear multiobjective problems implied by the Karush–Kuhn–Tucker necessary conditions are investigated. The regular case specific to nonscalarized MOPs is singled out when a nonlinear MOP with nonlinearities only in the constraints reduces to a nondegenerate linear system. It is shown that the trajectories of the Lagrange multipliers corresponding to the components of the vector cost function are orthogonal to the corresponding trajectories of the vector deviations in the balance space (to the balance set for Pareto solutions). Illustrative examples are presented.     

Nonscalarized multiobjective global optimization

A new approach to multiobjective optimization is presented which is made possible due to our ability to obtain full global optimal solutions. A distinctive feature of this approach is that a vector cost function is nonscalarized. The method provides a means for the solution of vector optimization problems with nonreconcilable objectives.This work was supported by the Natural Sciences and Engineering Research Council of Canada, Grant No. A3492.     

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.     

Sensitivity analysis on the priority of the objective functions in lexicographic multiple objective linear programs

This paper deals with a class of multiple objective linear programs (MOLP) called lexicographic multiple objective linear programs (LMOLP). In this paper, by providing an efficient algorithm which employs the preceding computations as well, it is shown how we can solve the LMOLP problem if the priority of the objective functions is changed. In fact, the proposed algorithm is a kind of sensitivity analysis on the priority of the objective functions in the LMOLP problems.     

Generalizations of a Theorem of Arrow, Barankin, and Blackwell in Topological Vector Spaces

It is proved that the density theorem of Arrow, Barankin, and Blackwell holds in a topological vector space equipped with a weakly closed convex cone to admit strictly positive continuous linear functionals. Moreover, several local versions of the Arrow, Barankin, and Blackwell theorem are given.     

Scalarizing vector optimization problems

A scalarization of vector optimization problems is proposed, where optimality is defined through convex cones. By varying the parameters of the scalar problem, it is possible to find all vector optima from the scalar ones. Moreover, it is shown that, under mild assumptions, the dependence is differentiable for smooth objective maps defined over reflexive Banach spaces. A sufficiency condition of optimality for a general mathematical programming problem is also given in the Appendix.     

集值向量优化问题ε-弱有效集的拓扑性质

本文研究集值向量优化问题ε-弱有效集的拓扑性质，证明了ε-弱有效解集的存在性、闭性、紧性和连通性。     

Solving combinatorial problems with combined Min-Max-Min-Sum objective and applications

The purpose of this paper is to introduce and study a new class of combinatorial optimization problems in which the objective function is the algebraic sum of a bottleneck cost function (Min-Max) and a linear cost function (Min-Sum). General algorithms for solving such problems are described and general complexity results are derived. A number of examples of application involving matchings, paths and cutsets, matroid bases, and matroid intersection problems are examined, and the general complexity results are specialized to each of them. The interest of these various problems comes in particular from their strong relation to other important and difficult combinatorial problems such as: weighted edge coloring of a graph; optimum weighted covering with matroid bases; optimum weighted partitioning with matroid intersections, etc. Another important area of application of the algorithms given in the paper is bicriterion analysis involving a Min-Max criterion and a Min-Sum one.     

Proper equality constraints and maximization of index vectors

In solving many practical problems, we have to deal with conflictive multiple objectives (in performance, cost, gain, or payoff, etc). Can all such objectives be achieved simultaneously? The general answer is negative. That is, most multiple-objective problems do not have supreme solutions that can satisfy all of the objectives. Many broader definitions of optimality like Pareto optimum, efficient point, noninferior point, etc, have been introduced in various contexts, so that most multiple-objective problems can have optimal solutions. But such optimal solutions do not in general yield unique vectors of optimal indexes of the conflictive multiple objectives. In most cases, we have to make appropriate tradeoffs, compromises, or choices, among those optimal solutions. To obtain the set of all such optimal solutions (in particular, the set of all optimal index vectors), say for a comprehensive study on appropriate tradeoffs, compromises, or choices, a usual practice is to optimize linear combinations of the multiple-objective functions for various weights. The success of such approach relies heavily on a certain directional convexity condition; in other words, if such convexity is absent, this method will fail to obtain essential subsets. The method of proper equality constraints (PEC), however, relies on no convexity condition at all, and through it we can obtain the entire set. In this paper, we attempt to lay the foundation for the method of PEC. We are mainly concerned with obtaining the set of all maximal index vectors, for most of the broader-sense optimal solutions are actually expressed in terms of maximal index vectors (Ref. 1). First, we introduce the notion of quasisupremal vector as a substantially equivalent substitute for, but a rather practical and useful extension of, the notion of maximal vector. Then, we propose and develop the method of PEC for computing the set of all quasisupremal (or maximal) index vectors. An illustrative example in the allocation of funds is given. One of the important conclusions is that optimizing the index of one objective with the indexes of all other objectives equated to some arbitrary constants may still result in inferior solutions. The sensitivity to variations in these constants are examined, and various tests for quasisupremality (maximality, or optimality) are derived in this paper.     

齐次多目标规划问题的KKT点

通过构造原问题的辅助问题,得到多目标规划问题的一些性质.并且给出目标函数是齐次函数的多目标优化问题KKT点的一个等价性质.     

Critical Points Index for Vector Functions and Vector Optimization

In this work, we study the critical points of vector functions from ℝ ⁿ to ℝ ^m with n≥m, following the definition introduced by Smale in the context of vector optimization. The local monotonicity properties of a vector function around a critical point which are invariant with respect to local coordinate changes are considered. We propose a classification of critical points through the introduction of a generalized Morse index for a critical point, consisting of a triplet of nonnegative integers. The proposed index is based on the sign of an appropriate invariant vector-valued second-order differential.     

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.