Exact generation of epsilon-efficient solutions in multiple objective programming

Abstract

Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. AbstractIt is a common characteristic of many multiple objective programming problems that the efficient solution set can only be identified in approximation: since this set often contains an infinite number of points, only a discrete representation can be computed, and due to numerical difficulties, each of these points itself might in general be only approximate to some efficient point. From among the various approximation concepts, this paper considers the notion of epsilon-efficiency which has also been shown to be of relevance other than merely for the purpose to approximate solutions.

Cone Characterizations of Approximate Solutions in Real Vector Optimization

“…As one such particular research question of interest and directly motivated by parallel methodological developments in Engau and Wiecek (2005a), we finally address the general problem of optimizing over the set of minimal or epsilon-minimal elements, or more precisely, the identification of (weakly) maximal elements among the set of (weakly) epsilon-minimal elements. Facing the challenge that for most practical situations the latter is not given explicitly and hence that actual optimization is in general not possible, we offer an alternative characterization of these maximal elements which as a by-product leads to an insightful result relating epsilon-minimal and weakly epsilon-minimal elements.…”

“…While Theorem 5.1 follows under quite general conditions, the second part requires some further assumptions, namely convexity and pointedness of the underlying cone C. An application of this result is given in Engau and Wiecek (2005a).…”

Tradeoff-based decomposition and decision-making in multiobjective programming

“…Also observe that the linear subproblem DL has four efficient single securities (6,10,11,18), including the three supporting points of the piecewise linear Pareto curve, whereas the nonlinear risk objective in RR leads to less obvious relationships and, in particular, produces only a single pure investment strategy that is Pareto optimal (15), but not efficient with respect to the specified domination cone D lr . Finally, we note that all but one (the second) pure investment strategies have a positive expected return in agreement with the generally favorable development of the European stock market between 2004 and 2007.…”

“…While Definition 2.2 was initially introduced as theoretical generalization of Pareto optimality by [22] and later adjusted for general cones, it is only recently that approximate efficiency has received increasing attention also for practical applications in multiobjective programming [including the author's work in [10,11], and other references therein]. This paper continues this previous work and further advances the use of epsilon-efficiency to model tradeoffs in the context of general domination cones and decomposition for MOP and MCDM.…”

Generating Epsilon-Efficient Solutions in Multiobjective Optimization by Genetic Algorithm

“…Afterwards, this concept has been used e.g. in [2] [4] [5]. To deal with a continuous multiobjective optimization problem, one has to consider a finite discretization of the set of feasible points (see Section 3 below).…”