Harold P. Benson

An improved definition of proper efficiency for vector maximization with respect to cones

Abstract Recently Borwein has proposed a definition for extending Geoffrions concept of proper efficiency to the vector maximization problem in which the domination cone S is any nontrivial, closed convex cone. However, when S is the non-negative orthant, solutions may exist which are proper according to Borweins definition but improper by Geoffrions definition. As a result, when S is the non-negative orthant, certain properties of proper efficiency as defined by Geoffrion do not hold under Borweins definition. To rectify this situation, we propose a definition of proper efficiency for the case when S is a nontrivial, closed convex cone which coincides with Geoffrions definition when S is the non-negative orthant. The proposed definition seems preferable to Borweins for developing a theory of proper efficiency for the case when S is a nontrivial, closed convex cone.

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.

An Outer Approximation Algorithm for Generating AllEfficient Extreme Points in the Outcome Set of a Multiple ObjectiveLinear 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.

Optimization over the efficient set

Abstract In this paper we study the problem of optimizing a linear function over the set of efficient solutions for a vector maximization problem. This problem arises whenever a linear function is available which acts as a criterion for measuring the importance of or for discriminating among the efficient alternatives that are available. Using the concepts of an efficient direction and of a direction of recession, we develop necessary and sufficient conditions for this problem to be unbounded. We also present necessary and sufficient conditions for efficient and for arbitrary solutions of the underlying vector maximization problem to be optimal solutions for this problem. Some of these conditions are geometric in nature, others algebraic. The algebraic conditions suggest potential computational procedures for finding an optimal solution to the problem. Finally, we give conditions under which the set of optimal solutions possesses certain special properties.

Concave Minimization: Theory, Applications and Algorithms

The purpose of this chapter is to present the essential elements of the theory, applications, and solution algorithms of concave minimization. Concave minimization problems seek to globally minimize real-valued concave functions over closed convex sets. As in other global optimization problems, in concave minimization problems there generally exist many local optima which are not global. Concave minimization problems are NP-hard. Even seemingly-simple cases can possess an exponential number of local minima. However, in spite of these difficulties, concave minimization problems are more tractable than general global optimization problems. This is because concave functions and minimizations display some special mathematical properties. Concave minimization problems also have a surprisingly-diverse range of direct and indirect applications. The special mathematical properties and diverse range of applications of concave minimization have motivated the construction of a rich and varied set of approaches for their solution. Three fundamental classes of solution approaches can be distinguished. These are the enumerative, successive approximation, and successive partitioning (branch and bound) approaches. After giving a brief introduction, the chapter describes and discusses some of the most important special mathematical properties of concave functions and of concave minimization problems. Following this, it describes several of the most important categories of direct and indirect applications of concave minimization. The chapter then describes each of the three fundamental solution approaches for concave minimization. For each approach, the essential elements are explained in a unified framework. Following this, for each approach, most of the well-known individual concave minimization algorithms that use the approach are explained and compared. The scope of the presentation of the solution approaches and algorithms is limited to those that use deterministic (rather than stochastic) methods. The chapter concludes with some suggested topics for further research.

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.

Towards finding global representations of the efficient set in multiple objective mathematical programming

We propose and justify the proposition that finding truly global representations of the efficient sets of multiple objective mathematical programs is a worthy goal. We summarize the essential elements of a general global shooting procedure that seeks such representations. This procedure illustrates the potential benefits to be gained from procedures for globally representing efficient sets in multiple objective mathematical programming.

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.

Using concave envelopes to globally solve the nonlinear sum of ratios problem

This article presents a branch and bound algorithm for globally solving the nonlinear sum of ratios problem (P). The algorithm works by globally solving a sum of ratios problem that is equivalent to problem (P). In the algorithm, upper bounds are computed by maximizing concave envelopes of a sum of ratios function over intersections of the feasible region of the equivalent problem with rectangular sets. The rectangular sets are systematically subdivided as the branch and bound search proceeds. Two versions of the algorithm, with convergence results, are presented. Computational advantages of these algorithms are indicated, and some computational results are given that were obtained by globally solving some sample problems with one of these algorithms.

Concave minimization via conical partitions and polyhedral outer approximation

An algorithm is proposed for globally minimizing a concave function over a compact convex set. This algorithm combines typical branch-and-bound elements like partitioning, bounding and deletion with suitably introduced cuts in such a way that the computationally most expensive subroutines of previous methods are avoided. In each step, essentially only few linear programming problems have to be solved. Some preliminary computational results are reported.